If 50g of hot water at 80degree C is poured into a cavity in a very
large block of ice at 0degree C, what will be the final temp. of the
water in the cavity?  How much ice must melt in order to cool the hot
water down to this temp.?

The first part is:0 °C.  Heat will continue to pass from the water to
the ice until they are in equilibrium.
For the second part, figure the required change in energy content of
the water and then calculate how much ice must melt to absorb this.