Following hydrogen depletion in the core, what event causes the core to contract and heat up again?

Reduced outward pressure from the slowing fusion reaction

The sustained equilibrium of a star is entirely dependent on the steady rate of hydrogen fusion providing sufficient outward pressure. When the hydrogen fuel in the core becomes largely exhausted and converted into helium ash, the rate of the primary fusion reaction naturally slows down due to the reduced fuel concentration. This decrease in energy output translates directly to a drop in the outward thermal and radiation pressure. With the outward push weakened, the overwhelming inward force of gravity once again becomes dominant, causing the core material to contract significantly. This contraction compresses the matter further, dramatically increasing the temperature, which eventually triggers the next phase of nucleosynthesis, such as helium fusion.