Pressure in a star
The relevant equations are dP/dr=-GM(r)*rho(r)/r^2, where dM(r)/dr=4*pi*r^2*rho(r) and M(r) is the total mass contained within radius r. So you can see immediately that dP/dr -> 0 as r -> 0. So there's no singularity at the origin. But to get the central pressure you have to integrate the M(r) first and then do the integral to get the pressure. I don't think you get get it by using only the total mass and properties at the center.