As someone already said in a similar thread (by you, I think), it would be unnatural for anyone to say "I have a father" or "I have the father". We would say "My father ..."
My father has leukemia.
My father is suffering from leukemia.
It is not possible to have more than one mother or father so "My xxx" is appropriate for both. This follows for direct relatives:
- My maternal grandmother has lupus.
- My maternal grandfather has flu.
- My paternal grandmother has Reynaud's disease.
- My paternal grandfather is in very good health.
- My daughter has tonsilitis.
- My eldest son has asthma.
With other relatives, we might use different constructions:
- One of my cousins has broken her leg.
- I have an aunt who used to live in Canada.
- My uncle owns a boat.
There is no "rule" for this. Even if you have ten uncles, you still might say "My uncle owns a boat" if it's really not important that the listener knows that you have more than one uncle, or which uncle it is. Be more specific only if you need to.