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Ken Veysey began his career as an apprentice with Arsenal, but on failing to make the grade followed his family who had moved to Devon. He joined Dawlish Town from where he was signed by Torquay United manager Cyril Knowles in November 1987. He waited a year to establish himself in the first-team, due to the form of first Kenny Allen and then Paul Crichton but took his chance when it came along, playing a major role in Torquay's run to Wembley in the 1989 Sherpa Van Trophy. However, on 15 April 1989 Veysey suffered a serious leg injury after a collision with his teammate David Cole in a 3-1 defeat by Leyton Orient at Brisbane Road. Kenny Allen was brought back to Plainmoor from semi-retirement and went on to play in the semi-final second leg victory over Wolves at Molineux (a game Wolves were so sure they'd win, they'd already booked their hotel for Wembley) and the final against Bolton Wanderers, with Veysey watching from the sidelines.
The following season he was an ever-present (totalling 55 appearances in all league and cup games) and was voted Player of the Year. After 72 league appearances for the Gulls, Veysey moved to Oxford United in October 1990, initially on loan and then making the move permanent for a fee of £110,000. He stayed at the Manor Ground for almost 3 years, making 57 league appearances before joining Reading on a free transfer in August 1993. He failed to break into the first team at Reading and subsequently returned to Devon, joining Exeter City on a free transfer in October 1993.
He made 12 league appearances for the Grecians before dropping out of league football in the summer of 1994, eventually joining Dorchester Town, who were then managed by former Torquay manager Stuart Morgan.
In August 1997 he was released by Dorchester after turning down a reduced contract and rejoined Torquay United, signing on a part-time basis, originally as cover for Matt Gregg. He returned to the first team in September 1997, playing in a 2-0 defeat against Swansea City at the Vetch Field, and although he was at fault for the second goal, retained his place in the side. As Torquay pushed for promotion, Veysey was an integral part of the side, but in the first leg of the promotion play-offs, handled the ball outside of his area, saving a certain goal, and was sent off and suspended for the play-off final at Wembley against Colchester United, Matt Gregg taking his place. The following seaosn, Veysey was injured, and Neville Southall was signed, originally as cover for Gregg, but soon became the first team regular and kept his place when Veysey had receovered from injury.
Veysey was released in the summer of 1999, having played 37 league games in his second spell with the Gulls. On 7 August 1999, he joined Kevin Hodges' Plymouth Argyle, as cover for the only Argyle keeper Jon Sheffield. He played 6 times in the league that season, but was released by Argyle in May 2000, joining Taunton Town during the close-season, although his appearances for Taunton were restricted by a back injury, and in October 2000 he was appointed goalkeeping coach at Torquay United.
In June 2007, Veysey again became Torquay United's goalkeeping coach.