Sometimes I will run a jumper cable from the negative post to a good ground on the block. If cranking improves it can mean the cable has a problem. Assuming the connection at the negative post is clean and tight. Somewhere there is some resistance. The starter may even be a lot hotter than it ought to be. Under most conditions, but not all, I would assume that the battery is at worst fair since it is able to heat the system up. But charging it and taking it to the auto parts is a free test.