Having had a quarrel with his wife,he left home in a bad temper.这个句子分词做状语,但是前面部分Having had a quarrel with his wife完全看不懂,究竟是什么跟什么啊,前面那个又不是从句,怎么会单独有一个动词had呢,又不是做谓语,为什么呢,

had a quarrel with his wife意思是和他太太吵架，had a quarrel with sb这是一个固定搭配意思是与某人争吵。having是动名词做主语是一个祈使句。

HAVING+动词的过去分词(现在完成时) 引导原因状语从句
正因为他跟他老婆吵了一架， 他心情很差的离开了家
前面是原因 后面是结果 have a quarrel with sb. 这个是固定短语

have a quarrel with 是固定短语 根据后面一句话用的是了left 用的是过去式 ,而前句和妻子吵架发生在前，用的是过去完成时had+动词的过去分词， 前句是he had had a quarrel with， 做状语 had要用ing形式

Having表伴随，做伴随状语，had a quarrel是个短语。

首先,跟他的妻子吵架,用词组,have a quarrel with his wife.
用现在完成时,则变成了have had a quarrel with his wife
现在需要把这句话变成现在分词做伴随状语的情况,把 have 变成 having ,就成了
Having had a quarrel with his wife