The core must resume its original collapse. This actually causes the luminosity to increase. The hydrogen burns in a shell around the helium core. The same mass is now in a much larger volume, so red giants have very low average density. Okay, well all this time the core continues to collapse and heat up, until finally it becomes hot enough to fuse helium into carbon by a reaction called the triple alpha process .
Carbon can also fuse with helium to form oxygen.
The star produces energy by burning helium in a shell around the core and hydrogen in a shell around that shell.