In the 1970's, Atkin and Swinnerton-Dyer conjectured that Fourier coefficients of holomorphic modular cusp forms on noncongruence subgroups of $\mathrm{SL}_2(\mathbb{Z})$ satisfy certain $p$-adic recurrence relations which are analogous to Hecke's recurrence relations for congruence subgroups. In 1985, this was proven in seminal work of Scholl and has recently been extended to the setting of weakly holomorphic modular forms by Kazalicki and Scholl. We show that these Atkin and Swinnerton-Dyer congruences extend to the setting of meromorphic modular forms and that the $p$-adic recurrence relations arise from Scholl's congruences in addition to a contribution of fibers of universal elliptic curves at the poles. Moreover, when the poles are located at CM points, we exploit the CM structure to reduce these $p$-adic recurrence relations to $2$-term relations. In this talk we will primarily illustrate these results through explicit examples. This is joint work with Ling Long and Hasan Saad.