Stuck at Holyrood with the task of ruling Scotland without him, Mary increasingly relied on her private secretary, the Italian Catholic, David Riccio.

结构剖析：这是一个含有过去分词短语的复合句。

主句是： Mary increasingly relied on her private secretary, the Italian Catholic, David Riccio，过去分词短语是：stuck at Holyrood with the task of ruling Scotland without him。

主句中的the Italian Catholic是做secretary的同位语，David Riccio是做the Italian Catholic的同位语。

过去分词短语中的非谓语动词stick与主语Mary存在被动关系，所以用过去分词，这个过去分词短语作时间状语。

在这个过去分词短语中，with the task of ruling Scotland without him是介词短语作原因状语，表示Mary要独自一人坚守在Holyrood的原因。

So Darnley is increasing estrangement from his wife, which gave the lords most offended by Riccio's access to the queen the opening they were looking for.

结构剖析：这个句子是含有定于从句的复合句。

主句是： Darnley is increasing estrangement from his wife，非限定性定语从句是：which gave the lords most offended by Riccio's access to the queen the opening they were looking for。

在非限定性定语从句中，引导词which是只带前面说到的情况（Darnley is increasing estrangement from his wife）。

在这个非限定性定语从句中，谓语的主干是gave the lords the opening。其余的most offended by Riccio's access to the queen是过去发分词作后置定语修饰名词lords，they were looking for是定语从句修饰名词opening，这个定语从句省去了引导词that或者是which。