  Dabholkar, Murthy and Zagier (DMZ) proved that there is a canonical
decomposition of a meromorphic Jacobi form of integral index for
$\mathrm{SL}(2, \mathbb{Z})$ with poles on torsion points
$z\in\mathbb{Q}\tau+\mathbb{Q}$ into polar and finite parts, and showed that
the finite part is a mock Jacobi form. In this paper we generalize the results
of DMZ to meromorphic Jacobi forms of rational index for congruence subgroups
of $\mathrm{SL}(2, \mathbb{Z})$. As an application, we establish that a large
class of single-centered black hole degeneracies in CHL models are given by the
Fourier coefficients of mock Jacobi forms. In this process we refine the result
of DMZ regarding the set of charges for which the single-centered black hole
degeneracies are given by a mock modular form. In particular, in the case
studied by DMZ, we present examples of charges for which the single-centered
degeneracies are not captured by the mock modular form of the expected index.
