这样的形容词短语(属现代英语中无动词分句的一种)到底在句子中作什么成分,与具体的句子语境有关。就你的句子来说:
Glad to accept, the boy nodded his agreement.
形容词短语(或叫无动词分句)Glad to accept 用作状语,男孩因为乐于接受,所以他点头表示同意(the boy nodded his agreement),即 Glad to accept 是因,the boy nodded his agreement 是果。
更详细的用法归纳请参见章振邦《新编高级英语语法》的相关说明: