married man with children
Refers specifically to a man who has a wife and children; often used to describe family responsibilities or settled life.
He has a wife and kids, so he can't just quit his job.
A married man with children has less free time than when he was single.
所帯持ち means a person who maintains a household, often implying marriage and possibly children, but is less specific than 妻子持ち.
Compound of 妻子 (wife and children) + 持ち (having, holding). The exact historical derivation is uncertain, but the word follows a common pattern for describing a person who possesses a certain family status.