Is it necessary to check for blood mixed with saliva upon repeated awakenings after Fajr prayer in Ramadan, knowing that it exits without feeling?

Intentionally swallowing blood invalidates the fast. What reaches the throat unintentionally or during sleep does not harm it. It is not obligatory to inspect the mouth every time after waking up, but whenever one is certain that blood has come out of the teeth, it is not permissible to swallow it.