For a normalized newform $g \in S_{k}(\Gamma_{0}(N))$ with complex multiplication by an imaginary quadratic field $K$, there is a mock modular form $F^{+}$ corresponding to $g$. K. Bringmann, P. Guerzhoy, and B. Kane modified $F^{+}$ to obtain the $p$-adic modular form by a certain $p$-adic constant $\alpha_{g}$. In addition, they showed that $\alpha_{g}=0$ if $p$ is split in $O_{K}$ and $p \nmid N$. On the other hand, the speaker showed that $\alpha_{g}$ is a $p$-adic unit for an inert prime $p\nmid 2N$ when $\dim_{\mathbb{C}} S_{k}(\Gamma_{0}(N))=1$. In this talk, the speaker determines the $p$-adic valuation of $\alpha_{g}$ for an inert prime $p$ under mild condition, when $g$ has weight 2 and rational Fourier coefficients.