Does he put his hand all the way into the mitt or does he leave part of it out? Reason I ask is if you put your hand all the way into the mitt then now he's catching the ball with part of his palm. He needs to keep part of his hand out of the mitt so that way the ball his landing in the pocket and it's his finger tips catching the ball instead of the palm.
Take a ball and put it in the palm of your glove hand - no glove. Now work a transfer into the finger tips of throwing hand. It's a jumbled mess. Yes there can be a transfer but it's awkward. Now do the same thing with the ball in the glove hand using the finger tips. It's going to be MUCH smoother transfer.
Couple of drills I use with my guys are
1. Three Fingers - I get about 6 or 7 feet away and toss a ball at them. They have to catch it with just the thumb, index and middle fingers while the ring and pinky stay extended. Don't let them tuck those two. Purpose - to get used to feeling the ball on the fingertips. You can also incorporate framing with this.
2. Ring around the rosie (I really need to find a cooler name than this) - you need at least two catchers (or catcher and coach) but three or more will work just fine. Let's go with four guys and you'll need two balls. Have two guys opposite of each other (they are standing in a box / square) with a ball in their left hand (no mitt) fingertips. When you say go they have to work a good transfer and just flip the ball to the next guy who catches with three fingers. Purpose - now both balls are going in a circle while working on catching with three fingers and getting a transfer to get rid of the ball. You can use two guys and two balls but it's pretty intense. Could lead to flipping instead of transferring.
3. Ball in Mitt - this is an individual drill and I incorporate the TEE drill for footwork with it. Paint your TEE on the ground and do whatever you want with this. While you're doing this each guy has a ball in his mitt but using the fingertips / hand partially out aspect. Have them squat and work coming out to throw while getting a transfer. Now they are in a throwing power position but if they do it correctly they won't fumble or search for the ball in the mitt. Purpose - work on all the different aspects of footwork involved with throwing while including transfers.
4. Rapid Fire - I pair catchers up and get them about 70 feet apart. One starts in the crouch with ball in hand like above and then throws to partner from him. He receives the throw in position 2 (runner on / higher up / ready to block or throw position) and throws back to his partner from the crouch. Once they throw they will need to hop back to try and to keep the distance from closing. Purpose - just another step added onto the drill above to incorporate the feet and transfer while adding a throw in the mix.
Couple of key thoughts -
1. Receive the ball as deep as possible to the body - a thrown ball is always faster than hands moving
2. Never listen to anyone who says it's like pulling the string back on a bow. It's nothing like that at all. It's about getting into a power position as quickly as possible.
Hope this helps some.