Looks like the Corsair PSU has two 6+2 pin PCI-E connectors, so that's all right for the 5870.
Sapphire recommends a minimum of 500W for the PSU for a single 5870, but not all PSUs are created equal. The Corsair HX450 is rated at 33A on its single +12V rail. Some cheap 600W supplies are rated lower than that.
I don't know what the actual power draws are for the 4850 and the 5870, but according to the calculation link above, the 5870 draws less than 10W more than a 4850.
I'd say that if the 4850 is OK, the 5870 should be too. Whether upgrading will improve your life, I can't say.