Remove any broken pieces from the block you wish to remove by breaking the pieces off at the attachment point with a chisel and hammer. Place the tip of the chisel at the damaged piece of block where it's still connected to the rest of the block. Hit the head of the chisel with the hammer to break it off.
Use a power drill with a masonry bit to drill a series of holes around the block in the mortar holding the block in place. Space the holes 1/4 inch apart and drill through the mortar as close to the rear of the block as possible.
Cut through the drilled holes with the chisel. Place the chisel point over the drilled holes covering the sections of mortar still intact in between. Strike the chisel head with the hammer to cut down through the mortar to the rear of the block all around the block perimeter. Clean out the broken mortar with the chisel and then remove the block from the wall.
Chip off any remaining mortar surrounding the hole where you took the block from with the chisel. Brush away the debris with a wire brush.
Mix mortar and water in a small bucket with a wooden stir stick until smooth and spreadable, with the consistency of peanut butter.
Use a spray bottle to wet down the blocks surrounding the hole. Spread mortar along the base of the hole with a trowel until it's about 1/4 inch higher than the mortar line on the surrounding blocks. Press the new block down into the mortar until the mortar is level with the other lines in the wall. Fill in the spaces around the block with mortar, using a tuckpointing trowel to push the mortar into the thin spaces on both sides and then along the top of the block.
Remove any mortar that squeezed from between the mortar lines onto the front of the blocks with a damp sponge. Allow the mortar to cure before touching the block. Cure times vary; consult the mortar instructions for the time for your specific mortar brand.