We study the properties of the fractional Poisson process with the Molchan-Golosov kernel. The kernel can be characterized as a compact since it is non-zero on compact interval. The integral of nonrandom function with respect to the centered and non-centered fractional Poisson processes with the Molchan-Golosov kernel is defined. The second moments of these integrals in terms of the norm of the integrand in L1/H([0, T]) space are obtained. Moment estimates for the higher moments of these integrals are established via the Bichteler-Jacod inequality.