Verified answer

Yes they can but...

if the doctor is a surgeon and is doing some urgent surgery he will not let it away to treat his child knowing the injury is not grave.

also he can treat his child during work in the hospital without problem but... one can say he is privileging his child in detriment of others.

and if, if and ifs

Obviously yes. Physicians can't practice medicine on immediate family members, though. The child would see another doctor or doctors in the hospital.