Increasing mass' effect on the balance between centripetal force and centrifugal force
The orbit of a planet is independant of the mass, as long as the mass is small compared to the star it's orbiting around i.e. it doesn't significantly change the centre of mass of the star-planet system. So changing the mass by an amount small compared to the mass of the star will make little difference.
Planets obviously do move their star a bit, because that's the way the first extrasolar planets were discovered i.e. by spotting that their star moved as the planet orbited around it. However this effect is pretty small unless we're talking about a Jupiter sized planet orbiting very near the star.
As long as you add the mass in a way that does not affect its speed, then the orbit is not changed(your star must be fixed as well).
Lets say the planet(mass $M$) is orbiting at a radius $R$, about a star of mass $M_\star$. The orbital velocity is $$v_1=\sqrt{\frac{GM_\star}{R}}$$. Now, in the comments you stated that you added the mass in a way that does not affect its velocity directly. Simce momentum is conserved, the only way to do this is to give the added mass $m$ a velocity $v_1$ as well at the time it reaches the planet. As you can see, when the planet captures the mass, there is no change in angular momentum ($mv_1R+Mv_1R=(m+M)v_1R$). Now, since theres no change in angular momentum, it will orbit at the same angular velocity. If its the same angular velocity, the radius is the same as well. So it stays in stable orbit. One can get this directly from $v_1=\sqrt{\frac{GM_\star}{R}}$ as well.
What if the mass was at rest and it was captured? Well, then by conservation of linear momentum, the velocity would decrease to $v_2$. Since the velocity decreased, it will go into an elliptical orbit. If the velocity had increased, the orbit could be elliptical, but it can be hyperbolic (greater than escape velocity) and leave the system as well. This depends upon the mass ratio.
If the central mass was not fixed, then the masses orbit around the center of mass(barycenter), and the orbital angular velocity is given by$\omega=\sqrt{\frac{G\mu}{R^3}}$ (note that I'm using angular velocity in this case, as the star and planet will have different velocities). $R$ is the distance between the objects, and $\mu=\frac{MM_\star}{M+M_\star}$ is the reduced mass. One can see that a whole variety of things can happen, depending on how you add the small mass, and on the ratio between the three masses. You may want to analyse this yourself (seeing as it's not part of the question and it's a pretty interesting exercise)