We study a class of curves which includes the hyperelliptic curves and the Fermat curves of prime degree. We compute their Hasse-Witt matrix when the curves are defined over an algebraically closed field of positive characteristic. In particular, we get a formula for the Hasse-Witt invariant of the Fermat curves at each prime not equal to the degree. These invariants depend only on residue degrees.
We show that there are infinitely many Fermat curves for which there exists a set of primes p with positive density such that the geometric fibre at p of the Fermat Jacobian is not isogenous to a power of a supersingular elliptic curve, but the Hasse-Witt invariant at p is equal to zero.