When cooking steak should the fat inside be cooked?

If there really is fat inside the steak, it is terribly undercooked. But this is unlikely to happen.

Animal fat is a saturated fat which melts somewhere around 35°C. If your steak hasn't reached 35°C internal temperature, it will be very visibly raw on the inside, with no apparent difference from the cold steak from the fridge. It will still be oozing blood-like liquid even after resting for over 10 minutes, and the liquid will be thicker than the juices of cooked steak. It will be deep red and slimy, and cold to the touch. Even the rarest steak you can buy is taken above 40°C.

But not everything white inside meat is fat. Connective tissue/fascia are white too. They are not really lump-shaped, but I don't know how precise your description is. Try cutting through one of the lumps. If it is tough and rubberlike, it is connective tissue (and if there is lots of it, your meat is not meant for roasting, but for a slow cooking method). If the knife goes easier through the lump than through the meat, it is fat, and your cooking method is seriously wrong.

In any case, the best way to know when steak is done is to use a meat thermometer. Stop when it shows around 61°C, the remaining heat transfer after removing from the heat will give you a medium steak.