  We report measurements of the branching fractions of singly
Cabibbo-suppressed decays $\Lambda_c^+ \to p \eta$ and $\Lambda_c^+ \to p
\pi^0$ using the full Belle data sample corresponding to an integrated
luminosity of 980.6 $\rm fb^{-1}$. The data were collected by the Belle
detector at the KEKB $e^{+}$$e^{-}$ asymmetric-energy collider. A clear
$\Lambda_c^+$ signal is seen in the invariant mass distribution of $p \eta$.
The signal yield of the $\Lambda_c^+ \to p \eta$ process is $7734 \pm 263$;
from this, we measure the ratio of branching fractions ${\cal B}(\Lambda_c^+
\to p \eta)/{\cal B}(\Lambda_c^+ \to p K^- \pi^+) = (2.258 \pm 0. 077(\rm stat.
) \pm 0.136(\rm syst. ))\times 10^{-2}$, from which we infer the branching
fraction ${\cal B}(\Lambda_c^+ \to p \eta) = (1.42 \pm 0.05(\rm stat.) \pm
0.11(\rm syst.)) \times 10^{-3}$. In addition, no significant signal for
$\Lambda_c^+ \to p \pi^0$ is found so an upper limit on the branching fraction
of ${\cal B}(\Lambda_c^+ \to p \pi^0)<8.0 \times 10^{-5}$ at 90\% credibility
level is set, more than three times better than the best current upper limit.
