Viet is a yakuza, but I can say for sure that he is not connected to that crime.
Viet is wondering if he should cut his connections with the yakuza and just focus on his work at Tofugu.
"He’s been married before and has three kids." "I know, but that has nothing to do with it."
A connection related to another connection obviously has some kind of connection, at least between all those meanings.
This is a jukugo word, which usually means on'yomi readings from the kanji. If you know the readings of your kanji you'll know how to read this as well.