Let $\pi:\mathcal{X}^{\prime}\to\mathcal{X}$ be an SNC model dominating the blow-up of $\mathcal{X}$ along $\mathfrak{a}_{m}$, so that we have $\mathfrak{a}_{m}\cdot\mathcal{O}_{\mathcal{X}^{\prime}}=\mathcal{O}_{\mathcal{X}^{\prime}}(-D)$ for some effective divisor $D\in\Div_{0}(\mathcal{X}^{\prime})$. By the projection formula we have

$$\omega_{\mathcal{X}}\otimes\mathcal{L}\otimes\mathcal{J}(\mathfrak{a}^{c})=\pi_{*}\left(\omega_{\mathcal{X}^{\prime}}\otimes\pi^{*}\mathcal{L}(-\lfloor c\,D\rfloor)\right).$$

Now $\pi^{*}\mathcal{L}-\tfrac{1}{m}D$ is nef and $c\,D-\lfloor c\,D\rfloor$ has coefficients in $[0,1[$. Lemma B.6 below together with the projection formula yields

$$R^{q}\pi_{*}\left(\omega_{\mathcal{X}^{\prime}}\otimes\pi^{*}\mathcal{L}\left(-\lfloor c\,D\rfloor\right)\right)=0\,\,\text{ for all }q\geq 1.$$

The Leray spectral sequence is thus degenerate and we conclude using Theorem B.3. ∎