It looks to me as if the hinges are either broken or simply loose / misadjusted. The doors don't look like they're broken, and it's unlikely the face frames of the cabinets have moved, so the problem is almost certainly with one or more hinges. There shouldn't be any rattle and slop in their movement. Look closely at the hinges and try to see where something is loose or out of place.

Most of the 35mm style hinges have 3-way adjustments. I can see some adjusting channels in your photo. Loosen the screws, readjust and tighten back up.

If your hinges can't adjust properly, you can buy new 35mm hinges that will fit perfectly with the existing door prep.
I can tell the ones on your cabinets are some of the cheapest- they do not appear to have much adjustment built in. But Bob's nephew is right, you can probably readjust these to get them to at least work for a baby latch.

Before buying new hinges, you need to measure the overlay. That is how much the hinged edge of the door covers the face of the cabinet when the door is closed. The hinges in your picture look like they may be 3/8" overlay, but I can't tell for sure.