Motherboard is clicking after installing a new graphics card

Okay

1) if your system is 4 years or older you should replace it, because the 'cost' to do upgrades as your doing as compared to 'replacing' the computer is MUCH MORE expensive (see below for my IMHO you should do now).

2) It sounds like there is potentially a short in either the Video port or the card itself causing the power to backlash over the board in 'pulses' and then feedbacks into the Integrated Sound.

3) The move to the new PSU was correct as the CARD REQUIRES 400W minimum, BUT you bought a really cheap one and MAY NOT ACTUALLY BE PROVIDING 400W power (I had a few posts on here reveal as low as 300W were actually what those 'cheapy' ones were doing and causing power issues like you said).

4) Steps to test this: Try the old video card in another computer, do you get the same results? If so bad card.
Try with a better (500W) powersupply, do you get the same problem still?
Unplug all components including the Mobo from the case, and place on wood (breadboard). Plug in just video card, CPU, Memory and Power to CASE POWER SWITCH, press on, do you get the same issue? If not then there is a short in the case causing the power to feedback, need a new case. If you still get the same issue then the mobo isn't handling the GPU and either a BIOS update or replacement of mobo is the solution.

5) As you can see to 'resolve' the issue will include more 'costs' (new case, new mobo, different video card, different PSU, etc.). Computers performance has radically increased and cost drastically descreased that PCs are disposable like cell phones (and cost as much, go compare it for yourself!). Your CHEAPEST solution would be to go to Walmart, pick up a i5Core desktop (or better yet a laptop) for only $349, then upgrade the PSU ($99+) to a decent power level for the VIDEO CARD your going to invest in ($129-$500). You will get a computer under warranty, that will last 5 years going forward (yours is obsolete and can't be pushed along any further), include a copy of Windows ($249 off the shelf cost normally) free, 6-8GB of DDR3 RAM, 500GB-1TB HDD, DVD/BlueRay, keyboard and mouse (sometimes even a FREE LCD monitor!!!) and a decent game card you can even play BF4 on High settings.

Last note: VISTA Hasn't been supported for years now, and has NUMEROUS issues and bad performance that the best solution was to upgrade to Windows 7 or stay back on XP. Even if you ignored all I said you still should dump your Vista, it will keep your system from performing like it was built to do.