Former Sheffield Wednesday defender Watson admitted he was under pressure after the Minstermen were knocked out of the FA Cup by lower-division Buxton and they followed it up with a 2-0 defeat at Leamington and a 0-0 draw with Curzon Ashton.
York are 11th in Conference North, a division they dropped into for the first time in 2017.
After the Curzon Ashton draw, York revealed "there has been a conversation with Steve and we have mutually agreed that it hasn’t worked out as anticipated and therefore, we need to part company."
His assistant Cummins has also left.
Watson joined from Gateshead in January 2019. York were second in what should have been his first full season in charge, but was abandoned in March because of Covid-19. Thanks to pressure from the Minstermen, the authorities agreed to play-offs but York were beaten in the semi-finals.
They finally moved into a long-awaited new stadium in Februrary, shortly before the 2020-21 season was also abandoned with the team eighth.
Askey is a former Macclesfield Town, Shrewsbury Town and Port Vale player who joined the coaching staff in November to support Watson "on a temporary basis".