$\phi_N$ which explicitly break the discrete symmetries but preserve the gauge symmetries of the standard model can be written as $$\begin{aligned}
\mathcal{L}_{\rm Planck} & \supset \frac{1}{M_{\rm Pl}} \bigg [ \sum _{i, \alpha, \beta} \phi_{Ni} (Y^{'}_1 \overline{L_{\alpha}} H e_{\beta} + Y^{'}_2 \overline{L_{\alpha}} \eta e_{\beta}+Y^{''}_1 \overline{L_{\alpha}} \tilde{H} \psi + Y^{''}_2 \overline{L_{\alpha}} \tilde{\eta} \psi) \nonumber \\
& +\sum_{i, \alpha, \beta} \phi_{Ni}(Y'_3 \overline{L_{\alpha}} \tilde{H} N_{\beta} +Y'_4 \overline{L_{\alpha}} \tilde{\eta} N_{\beta}) + \tilde{Y_1}H \eta^{\dagger} \psi \psi \bigg ] +{\rm h.c.}
\label{plH}\end{aligned}$$

We can also write down the dimension five terms simultaneously involving two of the $A_4$ triplet flavons $H_T, \phi_N$ which explicitly break the discrete symmetries but preserve the gauge symmetries of the standard model. They can be written as $$\begin{aligned}
\mathcal{L}_{\rm Planck} & \supset \frac{1}{M_{\rm Pl}} \bigg [ \sum _{i, j, \alpha, \beta} \phi_{Ni} (\tilde{Y^{'}}_1 \overline{L_{\alpha}} H_{Tj} e_{\beta} + \tilde{Y^{''}}_1 \overline{L_{\alpha}} \tilde{H}_{Tj} \psi +\tilde{Y'}_3 \overline{L_{\alpha}} \tilde{H}_{Tj} N_{\beta}) +\tilde{Y_2} H_{Tj} \eta^{\dagger} \psi \psi \bigg ] +{\rm h.c.}
\label{plHT}\end{aligned}$$

One can prevent the coupling of $\eta, \psi$ with standard model leptons at dimension five level by considering an additional $U(1)_X$ gauge symmetry under which $\psi$ is a Dirac fermion with charge $n_1$ while $\eta$ has a charge $2n_1$. Thus, $\eta$ can behave like a next to lightest 