We present the first search for heavy neutral leptons (HNL) decaying into $\nu e^+e^-$ or $\nu\pi^0$ final states in a liquid-argon time projection chamber using data collected with the MicroBooNE detector. The data were recorded synchronously with the NuMI neutrino beam from Fermilab's Main Injector corresponding to a total exposure of $7.01 \times 10^{20}$ protons on target. We set upper limits at the $90\%$ confidence level on the mixing parameter $\lvert U_{\mu 4}\rvert^2$ in the mass ranges $10\le m_{\rm HNL}\le 150$ MeV for the $\nu e^+e^-$ channel and $150\le m_{\rm HNL}\le 245$ MeV for the $\nu\pi^0$ channel, assuming $\lvert U_{e 4}\rvert^2 = \lvert U_{\tau 4}\rvert^2 = 0$. These limits represent the most stringent constraints in the mass range $35