What graphic card and processor?
When the PS is strong enogh you can use a 20pin one on 24pin MB.
If there is an aditional 4 pin connector on the motherboard you should have a fitting one on the powersupply too.
When you have a power connector on the graphics card it should be connected to.
About your question it can verry well be that the power supply is not strong enough, check its rating and think a moment how old it is.
It can also be that you have a heat problem. So open case and check fans for dust sitting in and below them, remove the dust. Check if they are turning fast, get some big ventilator and blow in air, try to start a game then.
Here at work i have powersupplies fail a lot but i think it is because we have laboratories and they mess up the power lines.