The energy radiated away is a very tiny fraction of the total mass energy, so g of the whole star would decrease very slightly. That has very little to do with what happens in the core. As hydrogen becomes helium the pressure produced by the hot ions decreases for the same temperature - four hydrogens have to fuse to make one helium. At the same temperature that means the same mass of helium exerts 1/4 the pressure of the same mass of hydrogen thanks to the ideal gas law:
PV = N.k.T
...N being the number of particles. To push back against the gravitational pressure of the star's mass, the core has to shrink and increase in temperature to maintain equilibrium as hydrogen becomes helium. Fusion reactions are very sensitive to the temperature and increase significantly as the temperature rises - but expanding the core too much causes the fusion reactions to slow down. Thus the core is always in balance between contraction and expansion, but slowly contracts as the total number of particles decreases slowly over time, and to maintain the pressure it has to slowly warm up. By decreasing in size, but increasing in temperature eventually a very hot, dense core results which causes the outer layers to puff-up and make the star expand
Expansion does happen after the core is depleted in hydrogen, but a shell of fusing hydrogen forms around the depleted core, so there's still lots of fusion going on. Eventually the core shrinks enough to allow fusion of helium, but the temperature must rise from its present 15,000,000 K to about 100,000,000 K before helium fusion becomes sustainable. Helium fusion is even more temperature sensitive than hydrogen fusion and in fact the Sun might undergo a runaway helium reaction that causes the core to explode... but not very much. This is known as the Helium Flash and it expands the helium burning core into a new, cooler configuration that allows the Helium Main Sequence to commence, with the Sun burning steadily at about 55 times its current output.
Fusing helium makes less energy than fusing hydrogen so the Helium Main Sequence only lasts 100 million years, unlike the 10 billion of the Hydrogen Main Sequence. If a star is heavy enough when the Helium Main Sequence ends the Carbon Main Sequence can kick in, but stars like the Sun are too low mass for that.