A cut on the paw is not necessarily a significant issue, but it is difficult to heal, & requires more time than you would think.
Make sure it is washed, dried & healthy tissue. If there is any sign of pus, bad smell, extreme redness or swelling, you DO need to follow up with a vet.
You mention that he reopens it when he jumps. This is the hard part. NO ACTIVITY!! I have Rhodesian Ridgebacks, I KNOW how difficult this can be. But, outside on a leash only, no running, & if he is very active in the house, crate him. One week of you taking him on walks vs. a large vet bill & complications from a reopened injury is worth it.
If you take him on walks, the pavement alone can open it back up. I would recommend some type of foot boot (you can get them @ the pet store, many different types, Polar Paws is one brand, usually sold in packs of 4, not the cheapest, but... ) In a pinch, you can wrap it in vet wrap (self-stick ace bandage), rolls are about $2 @ Fleet Farm.
Another route option is using a product like Tough Pad (I believe it is sold at places like Fleet Farm). It is a balm to put on the foot to toughen it up so that it is not as prone to cracking open.
Once you think it is healed, wait another week before resuming activities. A friend's dog ripped his pads on a running spree in a parking lot & she put him on crate rest for a week. Pulled him out & brought him to agility class, & he started refusing jumps... I suggested checking his paw & sure enough he had cracked it again (it wasn't bleeding yet). So they do take a while to heal, esp since the dog is constantly walking on them.
If you have a good relationship with your vet, you might want to call & just do a quick consult on their thoughts... let them know it's clean, not infected & you just need to get it healed... ask their thoughts regarding any type of wrapping (beyond what might be necessary for a walk) & application of products like Tough Pad.
*Disclaimer - I am NOT a vet. However, I am engaged in a number of dog activities that involve risk to injury, so I have gathered a lot of information from other dog enthusiasts & my vets.
T.