However, I would write "... the cancer ..." because he was suffering from a specific type of cancer. If he had many types of cancer and we don't know which of them caused his death we could write "... a cancer ...," but that would be unusual. If we were writing, for example, about Steve Jobs, we know his death was caused by cancer of the pancreas, so we would write "He succumbed to the cancer ..."
You say He succumbed to cancer, or He succumbed to pancreatic cancer.
But as soon as you add any defining relative clause, even one like yours that adds nothing which we don't already know, you have to add the definite article: He succumbed to the cancer he was suffering from.