the effective Dirac neutrino mass matrix can be written as (assuming $Y_{\nu 1}=Y_{\nu 2}=Y_{\nu 3}$) $$\begin{aligned}
M_D=M^0_{D}+M_{D}^1&=&\left(\begin{array}{ccc}
Y_{\nu 1}v_2& 0&0\\
0& 0 &Y_{\nu 1}v_2 \\
0 &Y_{\nu 1}v_2& 0
\end{array}\right)+\frac{u v_1}{M_{\rm Pl}}\left(\begin{array}{ccc}
(Y'_{3})_{11} & (Y'_{3})_{12} & (Y'_{3})_{13} \\
(Y'_{3})_{12} & (Y'_{3})_{13} &(Y'_{3})_{11} \\
(Y'_{3})_{13} &(Y'_{3})_{11} & (Y'_{3})_{12}
\end{array}\right),\nonumber\\
&=&v\left(\begin{array}{ccc}
y_1& y_2&y_3\\
y_2& y_3 &y_1 \\
y_3 &y_1& y_2
\end{array}\right), 
\label{mdirac1}\end{aligned}$$ where $y_1=Y_{\nu 1}+\frac{u }{M_{\rm Pl}}(Y'_{3})_{11}$, $y_2=\frac{u }{M_{\rm Pl}}(Y'_{3})_{12}$ and $y_3=\frac{u }{M_{\rm Pl}}(Y'_{3})_{13}$ with $v=v_2=v_1$.

Now, along with the Planck suppressed operator contribution in Dirac neutrino mass, the light neutrino mass originating from type I seesaw can be written as $$-M_{\nu}= M_D M^{-1}_{R} M_D^T. 
\label{mlight2}$$ After a rotation by $U_{\rm TBM}$, the light neutrino mass matrix can be written as $$\begin{aligned}
M'_{\nu}&=& U_{\rm TBM}^T M_{\nu} U_{\rm TBM}\nonumber\\
&=&\left(\begin{array}{ccc}
\frac{-v^2(ax_2+2bx_3)}{a^2-b^2}& 0&\frac{-\sqrt{3}v^2x_1}{a^2-b^2}\\
0& \frac{v^2x_4}{a} &0 \\
\frac{-\sqrt{3}v^2x_1}{a^2-b^2} &0& \frac{v^2(ax_2-2bx_3)}{a^2-b^2}
\end{array}\right). 
\label{mTB}\end{aligned}$$ where $$\begin{aligned}
 x_1&=&(2y_1-y_2-y_3)(y_2-y_3)\\
 x_2&=&-2y_1^2+y_2^2-4y_2y_3+y_3^2+2y_1(y_2+y_3)\\
 x_3&=&y_1^2+y_2^2-y_2y_3+y_3^2-y_1(y_2+y_3)\\
 x_4&=&(y_1+y_2+y_3)^2. \end{aligned}$$ Clearly an additional rotation in the 13 plane is required to diagonalise the above matrix via the relation $$U_{13}^{T}{M'_{\nu}}U_{13}={\rm diag} 
(m_1e^{i\gamma_1},m_2e^{i\gamma_2},m_3e^{i\gamma_3})$$ with $$\begin{aligned}
\label{u13}
U_{13}=\left(
\begin{array}{ccc}
 \cos\theta               & 0 & \sin\theta{e^{-i\psi}} \\
     0                    & 1 &            0 \\
 -\sin\theta{e^{i\psi}} & 0 &        \cos\theta
\end{array}
\right)  \end{aligned}$$ where $m_1, m_2, m_3$ are the real positive neutrino 