Corrente, 26, had 11 points, a plus-9 rating and 136 penalty minutes in 72 games with the Charlotte Checkers of the American Hockey League last season. He last played in the NHL in 2010-11, with six assists and a minus-5 rating in 22 games with the New Jersey Devils.
A 2006 first-round pick of the Devils (No. 30), Corrente has six points in 34 NHL games in parts of two seasons.
Kunyk, 24, made his NHL debut with the Lightning on April 13, playing 10:12 against the Washington Capitals.
Tampa Bay signed Kunyk as an undrafted free agent on March 20, after he completed his senior season at the University of Alaska-Fairbanks. He scored 60 goals and 133 points in 148 collegiate games, setting career highs as a senior with 22 goals, 21 assists and 43 points.