To give priority is to deal with one thing before all the others. The queue exists, and somebody has been moved up it.
It takes to, and there is no article — adding one is the usual error. The phrase always implies a queue and therefore somebody further back, so it carries a quiet decision about fairness, which is why hospitals, councils and rehousing schemes reach for it constantly. Take priority is the same fact from the other end: the urgent cases take it over the routine ones.
1phrasegive priorityTo deal with one thing before all the others.
“The clinic will give priority to the worst cases.”
give priority is an act of ranking.
Practised in Common Word Pairs