I get this a lot from new students in my shotgun training sessions. I focus on clays and wing shooting, but some of this transfers over to self defense loads and equipment.
Felt recoil has several deleterious effects. It can cause bruising on the arm, shoulder and face. It can result in physical anticipation of the shot, sometimes described as flinching. It can also adversely alter gun mount and other body mechanics. And it makes shooting a less than pleasant experience.
The first thing to check is gun mount and overall fit. The butt stock should rest squarely in the pocket where the deltoid and pectoral muscles come together. The stock should be brought up to the face, not the neck contorted to bring the face down to the stock. The pitch of the butt stock should be set so that the recoil is directed back into the shoulder, and not back and then up into the cheek. There are other aspects to gun fit, stance and mount that I can address in a different post if interest for that detail exists.
Second, make sure that your shotgun stock is not too short for your shooting stance and style. I see some students bring guns with very short LOPs, and they end up contorting their stance and mount to try and make the gun fit. Those contortions cause all types of bad habits, and can result in bruising on the face and the resultant raising of the head from the stock in anticipation of recoil and/or a thumb knuckle in the eye.
Next, look at the type of shells that you are using. If you are shooting a 12 gauge closed breach gun (pump, O/U, SxS, etc.), I suggest a 1 ounce load at 1150 fps or so for beginners. That is a nicely patterning, soft shooting load that will decisively break clays without the associated pounding of a heavier shot charge out of a faster shell. If you shoot an inertia or gas operated semi auto, you will have to try different loads to find the lightest recoiling one that will still consistently cycle the action. I find that a 1 pounce load in the neighborhood of 1200 fps is usually a minimum for those types of guns.
Finally, make sure that you have a quality recoil pad. Many factory pads are generic hard rubber, and provide very little protection. I personally recommend a nice soft Kick-EEZ pad, appropriately fitted for length of pull. Pachmayr Decelerator pads are also popular, but I prefer the energy dissipation of the Kick-EEZ. Softer pads do wear out easier, though, so be aware of that.