  In this paper we derive constraints on the emission of a massive
(pseudo)scalar $S$ from annihilation of neutrinos in the core of supernovae
through the dimension-4 coupling $\nu\nu S$, as well as the effective
dimension-5 operator $\frac{1}{\Lambda}(\nu\nu)(SS)$. While most of earlier
studies have focused on massless or ultralight scalars, our analysis involves
scalar with masses of order $\mathrm{eV- GeV}$ which can be copiously produced
during the explosion of supernovae, whose core temperature is generally of
order $T\sim \mathcal{O}(10)$ MeV. From the luminosity and deleptonization
arguments regarding the observation of SN1987A, we exclude a large range of
couplings $ 10^{-12} \lesssim {|g_{\alpha\beta}|}\lesssim 10^{-5}$ for the
dimension-4 case, depending on the neutrino flavours involved and the scalar
mass. In the case of dimension-5 operator, for a scalar mass from MeV to 100
MeV the coupling $h_{\alpha\beta}$ get constrained from $10^{-6}$ to $10^{-2}$,
with the cutoff scale explicitly set $\Lambda = 1$ TeV. We finally show that if
the neutrino burst of a nearby supernova explosion is detected by
Super-Kamiokande and IceCube, the constraints will be largely reinforced.
