  We apply heat kernel techniques in N=1 superspace to compute the one-loop
effective action to order $F^5$ for chiral superfields coupled to a non-Abelian
super Yang-Mills background. The results, when combined with those of
hep-th/0210146, yield the one-loop effective action to order $F^5$ for any N=2
super Yang-Mills theory coupled to matter hypermultiplets.
