My husband says he deleted the clash of clans game from his phone and doesn't play anymore, but I see he keeps getting the icon notifications on his phone. Is it possible or is he still playing? I have no idea about those things.

If the icon notifications are still appearing, the game is still active. However, you needn't worry about this. Clash of clans is not a game which requires you to persistently play it properly unlike a MMO which could require you to play a certain time. Clash of clans allows you to go on it time to time which doesn't seem any harm.

If you're really worried about your husband's "addiction", then try finding out if he has spent any real money on the game as you can buy gems with real money.