  We study the Standard Model effective field theory ($\nu$SMEFT) extended with
operators involving right-handed neutrinos, focussing on the regime where the
right-handed neutrinos decay promptly on collider scales to a photon and a
Standard Model neutrino. This scenario arises naturally for right-handed
neutrinos with masses of the order $m_N \sim 0.1 \dots 10\, \text{GeV}$. We
limit the relevant dimension-six operator coefficients using LEP and LHC
searches with photons and missing energy in the final state as well as pion and
tau decays. While bounds on new physics contributions are generally in the TeV
scale for order one operator coefficients, some coefficients, however, remain
very poorly constrained or even entirely evade bounds from current data.
Consequently, we identify such weakly constrained scenarios and propose new
searches for rare top and tau decays involving photons to probe potential new
physics in the $\nu$SMEFT parameter space. Our analysis highlights the
importance of performing dedicated searches for new rare tau and top decays.
