I can't tell you definitely if Thomas Cooper/Cowper is the son of Edmund, but I can provide a bit more information about them.
You should search the Suffolk RO catalogue:
There's a Thomas Cooper or Cowper of Mendlesham, turner, in the early 17th century:
1618. Anne Cooper, daughter of Thomas Cooper, placed apprentice by churchwardens and overseers to Miles Keble of Mendlesham, yeoman
1625. Margaret Cowper, daughter of William Cowper of Mendlesham, placed apprentice to Thomas Cowper of Mendlesham, turner, by churchwardens and overseers
And the Bedford catalogue:
There's a will of Michael Cowper of Henlow, 1539/40
He had several children, including Edmund, but it looks like Edmund was an adult, so too old to have a child in 1583.
Also one item in Common Pleas, Michael Cowper of Arlesey, Beds, probably the son of Michael Cowper of Henlow. The two are about a mile apart.
Hilary term 1543. f 424, first entry
The first defendant is Michael Cowper, of Arlesey, Beds, yeoman.