Not sure about United but, found this list from an 2012 article:
Middlesbrough played 10 games in April 2006.
Stockport County played 10 games in April 1997.
Arsenal played 10 games in April 1980.
Nottingham Forest played 10 games in April 1979.
Coventry played 10 games in April 1955.
Canvey Island apparently outdid them all:
Was a dead link to results list in
the article, apparently 12 league and cup games in April, and five matches on successive days at the beginning of May, including 9 games over the last 12 days of the season.
Also this from Mick Bodley, who was combining centre-half duties with his day job as a postman, he had to get up early for work each day having returned home from away trips long after midnight.