No, skaters cannot intentionally cover the puck or else it's a 2:00 delay of game penalty against the player.
If the puck is covered by a skater in the crease, a penalty shot is awarded to the opposing team. The crease is the painted area in front of the goal, and that area extends upward from the ice surface up to infinity. (Think of a half cylinder extending upwards from the ice.) If a defending skater cups his hand around the puck or otherwise covers it while in this "half cylinder" a penalty shot is awarded. The puck does not have to be touching the ice.
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Not with his hand if the goalie is in the crease.
However, a player may fall (lay) on his goaltender, inadvertently covering the puck with his body.
You can come out to the face-off dots in the zone you are in to cover it up, beyond there, you must play it or else it will be a delay of game penalty.
You are not allowed to hit a goalie in the NHL. Of you do you will get a 2 minute penalty for goalie interference.
Yes, but only if the other player is inside his crease (the blue semi-circle in front of the net)