A youth team product, Drewe made his debut in March 1997 against Crystal Palace as substitute. By the end of 1996/97, he had made 8 appearances (5 as substitute) and had scored his first goal. However he fell out of favour during 1997/98 and made just 1 further appearance.

Drewe was placed on the transfer list on October 7th 1998 by Bruce Rioch. Peterborough manager Barry Fry made an offer for Broughton but it was considered derisory by Bruce Rioch. Brentford also expressed an interest in signing Broughton and Drewe eventually completed his move to Griffin Park on 28th October for the initial fee of £15,000 with three future payments of £20,000 dependent on appearances. After just three weeks with Brentford, playing only 56 minutes, Drewe was transferred to Peterborough for £15,000 following Brentford's signing of Leo Fortune West from Lincoln. There is confusion over whether Brentford supremo Ron Noades decided that Drewe was not good enough, replacing him with Fortune-West or whether Drewe sought an exit rather than fight for his first team place.

Drewe became a regular in the Peterborough attack and has scored a couple of times for them including the winner at Leyton Orient on 12th December 1998. He finished the 98/99 season with 8 goals from 27 appearances but could not establish himself in the first team in 1999/2000. Jason Lee arrived at London Road in January, on loan from Chesterfield with the prospect of a permanent move. To fund a longer stay for Jason, manager Barry Fry sought to reduce the club's wage bill. Despite interest from Division Three strugglers Chester City, Drewe joined Conference side Nuneaton Borough on loan on 7th February 2000. This move was extended until the end of the 1999/2000 season enabling Peterborough to bring in new players for their promotion challenge.

Following Peterborough's promotion, Drewe agreed to go on the transfer list and find a new club for the 2000/01 season. Both Nuneaton and Morecambe had unsuitable bids rejected. On 21st September 2000, Drewe joined Conference side Dagenham & Redbridge on loan. However he was unwilling to drop permanently into the Conference and in mid October 2000 was having talks with Leyton Orient. Drewe failed to agree terms with Orient and was later the subject of a £50,000 bid from ambitious Conference side Telford. Returning to London Road in December 2000 having failed to agree terms with either Orient or Telford, Drewe headed out on loan again, this time to Conference side Stevenage Borough for whom he scored twice on his debut. After a month had passed and the loan expired, Drewe returned to London Road only to be immediately linked with another Conference side, this time Yeovil who were keen on him and Posh team mate Ritchie Hanlon. Despite a trial at Reading where he appeared for their Reserves, and interest from Rushden & Diamonds, Drewe joined Kidderminster on loan in January 2001 which led to a £50,000 transfer the following month as Drewe impressed Jan Molby, the Harrier's boss, with 2 goals in his first 2 appearances for the Harriers.

In September 2001, Oxford United manager Mark Wright suggested that a move for Drewe was a possibility as he was the type of player he was looking for. Drewe however stayed at Kidderminster but fell out of favour at Christmas 2002. At the beginning of June 2003, Southend manager Steve Wignall revealed that he had been in positive discussions with Drewe to bring him to Roots Hall. Harriers were unlikely to object having given Drewe a free transfer. This materialised on 11th June with Drewe signing a two year deal with the Shrimpers.

On October 14th 2002, Drewe joined Rushden and Diamonds on a one month's loan, making his debut on the 16th October in a 1-0 home defeat to Chester City. This was extended to a second month as Drewe did well. With five goals in ten matches, Diamonds would have liked to extended his stay in Northamptonshire and Southend were willing to let him do so with Fredy Eastwood, Wayne Gray, Lawrie Dudfield and Tes Bramble all ahead of him in the Shrimpers' attack. However Rushden's finances would not allow it. "We offered Drewe to Rushden for another month on loan but apparently they can't afford him at the moment so he will come back to Southend," said Southend manager Steve Tilson to Southend's website.

Soon after his return to Southend,he joined Wycombe Wanderers on a two month loan deal. He returned to Rushden & Diamonds in mid-February 2005 signing a contract until the end of the 2004/5 season.

A run of goalscoring form in the spring of 2006 saw the News of the World speculate on April 23rd 2006 that Drewe was a target for Championship sides Brighton, Crystal Palace and QPR. Following the Diamonds' relegation to the Conference at the end of the 2005/ season, Drewe opted to leave the club. He had scored 10 goals in 37 games for Rushden last season, but also collected two red and nine yellow cards. On 16th June 2006, he signed for Chester City on a two year contract.

On 25th October 2006, it was reported that new Macclesfield Town boss Paul Ince could make Chester City striker Drewe Broughton his first signing since taking over at the Moss Rose. According to Chester, Ince enquired about the availability of 28-year-old Broughton, who is also interesting fellow strugglers Boston United. City chairman Stephen Vaughan said: "Drewe hasn't figured regularly in our starting line-up recently. A loan may be beneficial to both parties." Drewe ended up joining Boston on loan on 27th October 2006. Drew scored 8 goals in 26 appearances for Boston but wasn't able to prevent them dropping out of the league. In May 2007, Chester reached a settlement with Broughton to allow him to leave the Deva Stadium

In July 2007, Drew trialled with Bury and was one of the shining lights in the 2-1 pre-season friendly defeat at Altrincham. He then trialled with Milton Keynes Dons who signed him on a one year contract on 24th July 2007. Although the Dons sit proudly at the top of the table as of 25th January 2008, Broughton has not been a regular in the side with most of his 13 appearances coming as substitute. On 25 January, he was loaned to Wrexham who were then bottom of League Two. He made his debut the following day in a 1-1 draw against Bradford. Alongside Broughton in the Wrexham line-up was Chris Llewellyn whilst Bradford fielded new signing Paul Heckingbottom in their team.

Drewe returned to the MK Dons in April 2008, having been unable to assist Wrexham avoid relegation in his sixteen appearances for the Welsh side. On 17 May 2008, Drewe was one of five first team squad members released by the MK Dons following their promotion to League One, the others being Gareth Edds, Matt Carbon, Sam Collins and Kieran Murphy. Drewe had made 19 appearances for the Dons but 13 of these had been as substitute.

Drewe joined Gillingham for pre-season training in July 2008 but was not taken on. He trialled with both Luton and Rotherham during July and was invited back or another trial with The Millers in August 2008. He was offered a short term contract by Millers' boss Mark Robins which was extended to a one year contract on 28th August. Robins said "Over the last couple of games he has done enough to earn a longer deal and he gives us something different."

On 16th February Drewe joined struggling League Two rivals Lincoln City, managed by Chris Sutton, on a one month emergency loan. He had scored 12 goals in 51 appearances at the Don Valley Stadium before being dropped by Robins' replacement Ronnie Moore in January 2010. Although Broughton failed to score in seven appearances, he was recalled by Rotherham to assist their promotion ambitions. Following their play-off final defeat, Broughton was released and signed a one year contract with Lincoln on 2nd June 2010

On 18th February 2011, Drewe was loaned to AFC Wimbledon until the end of the 2010/11 season. He had made 20 appearances for the Imps during 2010/11 but had not scored. He made his debut for Wimbledon on 19th February 2011 and scored against Tamworth. He made 8 appearances, scoring twice for AFC Wimbledon before returning to Sincil Bank

On 7th May 2011, Lincoln were relegated to the Conference and four days later, he was released along with all bar three of the first team squad. After being released, Drewe went on trial at Luton scoring two goals on debut, in a 4-2 victory over Hitchin in a pre-season friendly.

On 8th September 2011, Drewe signed for Conference newcomers Alfreton Town on non-contract terms. He made his debut on 17th September, scoring in a 3-1 defeat away at Kidderminster Harriers.

He departed Alfreton in early October after receiving a red card for violent conduct in the 5-2 defeat to Grimsby on 1st October 2011. He was subsequently in talks with Kings Lynn Town but instead joined Thurrock on 21st October 2011.

He moved on to Arlesey Town of the Southern League Premier Division, making his debut on 28th January 2012. After two appearances and one goal, he signed for Darlington in March 2012, making his debut against Stockport on 3rd March 2012. He made 11 appearances for Darlington, scoring once but was released at the end of the season.