The survey found that government and public administration employees were the most likely to wake up thinking about their careers.

我认为，原句是在likely 之后漏掉了to，因为英语有这样一个句式：be likely to do sth. 有可能做某事。

thinking about their careers，为现在分词短语，作wake up 的伴随状语。

【翻译】调查发现，政府和公共行政部门的员工，最有可能的是，醒来后就考虑自己的前程如何。