My sons PSP 2004 keeps asking do you want to quit this game while he is playing. He is not thouching the home button. Can somebody please help?

The problem, as you describe, is exactly that! One of your buttons is stuck down in the pressed position. If you press Home and the screen comes up asking you if you want to quit and a button is already pressed down, you will not be able to press X to confirm yes or no.

You didn't specify which PSP you have, 1001, 2001, or 3001, but the following website sells the part you need to fix this issue. They also perform repairs if you're not comfortable doing it yourself. It's a pretty cheap and straightforward part to replace:

PSP 1001:

http://store.richspsxparts.com/pspbucibo.html

PSP 2001:

http://store.richspsxparts.com/pspslpricibo.html

PSP 3001:

http://store.richspsxparts.com/pspslpspsric.html

the thing is that ur home button is stuck wen ur playing a game if u press down the home button it will ask u that

To correct this problem you need to update the firmware of your PSP so this problem happen again......

Maybe because it is highly sensitive type.Even if you touch it with a little force while playing this can happen.

Your PSP needs to be fixed. The home button might be stuck. No software solution for this, it's hardware related.

maybe you keep pressing the home button

This shouldn't happen. This is the first time I've heard about a problem like this. My instincts tell me that it is probably the home button. it may look like there's nothing wrong with the button on the outside but underneath the home button. Don't try to fix it yourself, try to find a professional repair shop &get it fixed there.

check the home button it could be jammed

it could be that someone pushed the button in too hard and it wont pop back up because even if you lightly touch it, it could go to that screen

This may be a number of problems. The disc might be scratched or damaged. Or there could be dust on the leans.

You say it's a second hand PSP? In that case, you may need to repare it. I'm not quite sure where to go to get repares. Sorry.