and criticizing her time as secretary of state as the tycoon looked to reset his presidential campaign after several rocky weeks.

1. 第一个 as 做介词，引导介词短语做 her time 定语。——你的理解是对的。as secretary of state 作定语修饰her time，意思是“作为国务卿任职期间的表现”。

2. 第二个 as 是做连词后面跟句子还是介词？后面跟名词 tycoon，之后是that省略了的定从吗？因为感觉翻译过来做定从更合适。——第二个as 是连词，表示“原因”。as the tycoon looked to reset his presidential campaign after several rocky weeks.

look to do sth. 属非正式英语。意为“计划/期待/盼望做某事”。