HARTFORD, Conn.—Hillary Clinton’s support for stronger gun-control laws has become the central theme of her bid to win the Democratic primary in Connecticut, the state where a deadly shooting at an elementary school sparked a national dialogue about firearms.

In her first public campaign stop in the state this year, Mrs. Clinton spoke Thursday at a Hartford YMCA about gun policy. The event included two people who lost family members when a gunman fatally shot 20 children and six adult staff members at Sandy Hook Elementary...