The most obvious thing that comes to mind since you say the hinge is close to the power button and you can make the power button work by moving the hinge is that you have a loose connection (wire) on the power switch that gets pushed around by the hinge.  You'd have to open the case to inspect, but it sounds logical.

>I bought a second hand Gateway Solo 9300 with a 15.7" screen. I sometimes
>have trouble turning it on. It seems as if the power switch is not closing
>when I press the start button. At first, I attributed it to a bad battery,
>but I have a new battery and the problem continues.
>
>After two months of use, I have noticed that the operation of the power
>switch is somehow affected by the movement or position of the screen/cover.
>When the switch does not work, I can make it work by moving the screen
>forward or back. The situation is the same if I try to shut down using the
>switch. I notice that the right hinge is very close to the start button.
>
>Does anybody know why this would be happening, and what I should or should
>not do about it? I can live with it if it doesn't get worse.
>
>

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