Cut a little beveled/angled piece out of the old board and fill in the groove with joint compound, then embed a paper tape quite broadly but thinly.
Since the joint compound groove is missing on that board, you need to cut out a bit, bevel it with about a 45-degree angle, slightly more than half of the board's thickness.
After you've applied joint compound and sanded it to your satisfaction, apply primer and use a raking light to add touch-up compound.
It might be good to check if you need to add screws to the existing board so it doesn't flex.
Since the joint compound groove is missing on that board, you need to cut out a bit, bevel it with about a 45-degree angle, slightly more than half of the board's thickness.
After you've applied joint compound and sanded it to your satisfaction, apply primer and use a raking light to add touch-up compound.
It might be good to check if you need to add screws to the existing board so it doesn't flex.