Procedures about treating/not treating during the round are variable. In my place the attitude is liberal, that is: the team can chose to treat the cut on their own BUT if they have no medic/paramedic they must use only external use material (adrenalin and serum or other things that can be placed on the cut); the stuff must be examined and approved by the event's physician prior to the fight.
Adrenalin is usually 1/1000 and that is too strong for the external use, it should be diluted 1/10, that is, to reach 1/10000 concentration with appropriate solutions (sodium chloride 0.9%, Ringer solution). It has a strong vasoconstrictor effect that initiates almost immediately.
If adrenalin is "prescription only", try to obtain fibrin sponges. Fibrin sponge preparation contain the actual substance of the blood clot prepared so as to exclude any germ contamination and can be moisturized with a saline sterile solution and applied on the spot. Take care it turns into a paste when moistened. It is delivered in pieces of ~4 / 2 inches and should be cut in smaller pieces, applied on a moist sterile bandage sheet and placed on the cut. It shortens the clot formation because from the first blood drip up to fibrin precipitation normal chemistry of the blood needs a lag time.
Calcium considerably accelerates the coagulation, so any sterile calcium solution can mend the cut faster.
The good combination is adrenalin 1/10000 plus a few drops of calcium chloride 1% used to moisten a fibrin sponge.

Example of fibrin sponges commercial form:
http://www.magazinuldentar.ro/index.php?main_page=product_info&cPath=35&products_id=215

There are also OTC (over the counter) kits containing various mixes but you should check the content. Dentists can help because they use such things after a tooth extraction. An example:

http://www.magazinuldentar.ro/index.php?main_page=product_info&cPath=35&products_id=215