  In a private communication, K. Ono conjectured that any mock theta function
of weight 1/2 or 3/2 can be congruent modulo a prime $p$ to a weakly
holomorphic modular form for just a few values of $p$. In this paper we
describe when such a congruence occurs. More precisely we show that it depends
on the $p$-adic valuation of the mock theta function itself. In order to do so,
we construct a family of mock theta functions in terms of derivatives of the
Appell sum, which have a special Fourier expansion at infinity.
