With extra income from private sponsorship, the club is just about managing to keep its head above water.

keep 是动词，意思是"保持、维持"。 its 是物主代词，指代前面提到的"俱乐部"。 head 本义是"头"，这里比喻为"主要部分"或"核心"。 above 是介词，表示"在...之上"。 water 本义是"水"，这里比喻为"困境"或"财务危机"。 keep one's head above water 是一个习语，字面意思是"让头保持在水面之上"，比喻"勉强维持运营"或"避免破产"。