Hello Shen, Someone's connections are the people who they know or are related to, especially when they are in a position to help them.( Notice that "connections" here is a plural noun). Having important connections is being related or befriended in a way or the other with key personalities in a certain society ,who have some power or authority to help achieve someone's goals or give them support in any matter. In your sentence Jonathan's family is related to powerful ,influential or affluent people in the society, so despite their little wealth their contacts remained an asset for them.

It means he didn't have money, but he had relationships with influential other people. This might be thru military connections (he served in a war with someone important) or political (he was friends with the town mayor) or in business (he is friends with the businessman who owns the largest industry in the town) or in religion (he is friends with the bishop) or something similar.

Hello Shen In this sentence "connections" means "relations" with some important members of the society.

"Connections" in this usage mean those that one knows that have influence in society. In the sentence the phrase means that the family knew others that were wealthy, influential and/or powerful members of the society.