  Nucleon weak matrix elements can be extracted from nucleon correlation
functions with lattice QCD simulations. The signal-to-noise ratio prohibits the
analysis at large source-sink separations and as a consequence, excited state
contamination affects the extraction of the nucleon matrix elements. Chiral
perturbation theory (ChPT) suggests that the dominant contamination in some of
these channels is due to $N\pi$ states where the pion carries the same momentum
of the current. In this talk, we report updates on the variational analysis
with $qqq$-operators (nucleon-like) and $(qqq)(\bar{q}q)$-operators
(nucleon-pion-like) where we report for the first time some preliminary results
of $\langle N\pi| \mathcal{J}| N \rangle $, modulo some kinematic and volume
factors, and we compare the results against ChPT. This pilot study is performed
on a CLS ensemble with $N_f=3$, $m_\pi \approx 420~\mathrm{MeV}$, $a\approx
0.1~\mathrm{fm}$ and $T=2L\approx 4.8~\mathrm{fm}$.
