You will find it worthwhile to keep ____ good terms withthe supervisor of your department.

c

keep on good (friendly) terms with sb.
与某人保持良好(友谊)关系, 交朋友

keep on

继续前进; 继续工作
Turn left at the corner and keep on as far as the church.
到街角向左拐, 一直向前走到教堂。
继续穿着〔戴着〕
It's rude for a man to keep his hat on indoors.
男人进屋不脱帽是不礼貌的。
继续雇用; 继续使用或占有
Are you able to keep both the gardeners on?
你还能继续雇用这两名花匠吗?
反复提及; 抱怨
The old lady kept on about her illness.
那老太太没完没了地讲她的病情。
(使)继续朝…行进
If we keep on our way for another hour, we should reach the village.
如果再继续向前走一小时, 我们就到那个村子了。
(使)继续服药〔接受训练〕
Will you keep on the course of exercises until you are thinner?
你要继续这项训练直到身体瘦下来为止吗?

keep on 固定词组，保持的意思