I wouldn't give Hart Trophy voting all that much credence, but given that the very same reporters and columnists who voted for the awards are also telling me that Ilya Kovalchuk is the 'prize of free agency', I thought we should look at how they've voted for the NHL's MVP since the lockout:
During a period where he scored 230 goals, only a handful of voters thought that Kovalchuk was one of the top five players in the league in any given season. The average salary for the players in this group who are are out of their entry-level period but under 35 and have contracts for next season is $7.1M, which roughly corresponds to being a 3-win player. (The bottom 20 players receiving votes averaged $6.2M.)
It's not clear that Kovalchuk is a 3-win player. He's in the ballpark, but you, the voters, took the under, estimating his value at $6.4M - 2.5 wins - for the 39th-highest salary overall in the league. Would Ilya Kovalchuk sign a contract that would pay him over $1M less than he made this season? I'd assume he wants to be making $8M or more.
So according to both you and the Hart voters, any team that acquires Kovalchuk will be overpaying for him, whether it's a long-term deal or a very expensive one-year contract. Kovalchuk's PP talent and shooting skill make him a very good player, and he's the free agent who can make the single-largest improvement in your team, but in a world ruled by the salary cap, he is decidedly not 'the prize of free agency.'