We have searched for the Cabibbo-suppressed decay $\Lambda_c^+\to\phi p\pi^0$ in $e^+e^-$ collisions using a data sample corresponding to an integrated luminosity of 915 $\rm fb^{-1}$. The data were collected by the Belle experiment at the KEKB $e^+e^-$ asymmetric-energy collider running at or near the $\Upsilon(4S)$ and $\Upsilon(5S)$ resonances. No significant signal is observed, and we set an upper limit on the branching fraction of $\mathcal{B}(\Lambda_c^+\to \phi p\pi^0) <15.3\times10^{-5}$ at 90% confidence level. The contribution for nonresonant $\Lambda_c^+\to K^+K^- p\pi^0$ decays is found to be consistent with zero and the corresponding upper limit on its branching fraction is set to be $\mathcal{B}(\Lambda_c^+\to K^+K^-p\pi^0)_{\rm NR} <6.3\times10^{-5} $ at 90% confidence level. We also measure the branching fraction for the Cabibbo-favored decay $\Lambda_c^+\to K^-\pi^+p\pi^0$; the result is $\mathcal{B}(\Lambda_c^+\to K^-\pi^+p\pi^0)= (4.42\pm0.05\, (\rm stat.) \pm 0.12\, (\rm syst.) \pm 0.16\, (\mathcal{B}_{\rm Norm}))\%$, which is the most precise measurement to date. Finally, we have searched for an intermediate hidden-strangeness pentaquark decay $P^+_s\to\phi p$. We see no evidence for this intermediate decay and set an upper limit on the product branching fraction of ${\cal B}(\Lambda_c^+\to P^+_s \pi^0)\times {\cal B}(P^+_s\to\phi p) <8.3\times 10^{-5}$ at 90% confidence level.
