A Bug's Life

I've done mostly carpentry, not cabinet work, so you should ask and get more opinions, but here goes:

The "base of a sleeve" looks like it was subjected to some side force which snapped it along the grain. If the pieces fit together well (no gaps) they are a good candidate for glueing with a regular wood glue ("Elmer's Glue-All" in the US). Apply a very thin layer and hold the pieces together firmly for 15 minutes (this is where a clamp is nice). If there are gaps you can glue the pieces back together with epoxy which can fill voids and still is fairly strong. You may need to back the piece with a thin piece of wood to strengthen it and hold the pieces in place. You can get very thin sheets of wood at a hardware or art supply store. Cut with a sharp knife ("X-acto" or box cutter). Fasten to the back of the piece with glue and little staples.

The horizontal bit looks like the wood is damaged, not just broken. If the pieces fit together perfectly, great, you can glue them. Otherwise you will need to replace the damaged wood and/or add new wood to strengthen it. It looks like the piece is load bearing, but not in a major way, otherwise they would not cut that big notch into it. Also it looks like it is not normally visible when assembled. I would probably get some small pieces of hardwood (oak is good) that are about the same size as the piece being repaired, and screw them on to each side. The reinforcing pieces may need to be cut down to fit around other other pieces under the table. (Such as for whatever reason that notch is there.) The screw holes should be pre-drilled, to avoid splitting the wood. The screws should be counter-sunk for tidiness (no heads sticking out). They should be about 2.75 cm in length, so they go through the piece in the middle and into the reinforcing piece on the other side. Use good quality brass screws so they can't rust. If there isn't room for reinforcing pieces, another fix would be to cut out the damaged wood and replace it with new. I think the way they do that is to cut at a 45° angle so the pieces can be screwed together from both the top and the bottom.