There are two versions of the 7750, one with a stock clock of 800 MHz and one with a 900 MHz clock. There is no difference between the naming of the two, you simply have to look. However, to get to 900 MHz, AMD did have to a 6-pin auxiliary power connector. So, no, the 800 MHz versions do not need and additional power, but the 900 MHz versions do.
As for the 375 watt PSU, you can run almost any card with that mount of power, so it will certainly handle a 7750.