I experience again the deep-felt wish to be part of a married couple, to sit by the fire in winter with the man who is my husband. (Para. 1)
Meaning: Again I deeply feel the strong desire to be part of a married couple, to sit by the fire in winter with the man who is my husband.
Please notice that the infinitive phrases — "to be part of a married couple, to sit by the fire in winter with the man who is my husband" — modify the word "wish".
experience: 1) vt. feel, live through
She experienced pain in thinking of her husband who had passed away years ago. 她丈夫于几年前去世， 她经历了一段思念亡夫的悲痛过程。
2) n. (i) (uncountable) knowledge gained by practice
He has plenty of teaching experience. 他很有教学经验。
(ii) (countable) something that happens to
Travelling experiences by bike are quite exciting. 骑自行车旅行是一种令人兴奋的体验。
a part of: one of the pieces, sections or segments that it is made of
She lived there for the greater part of her life. 她一生中大部分时间都住在那里。
Respect is a very important part of any relationship. 相互尊重是人际关系中一个重要的方面。