I would ignore the answer about GJK.
Since you can detect if the point is in your polygon, all you need to do find which face is the axis of minimum penetration. This tells you what direction to press your two objects apart. This direction is called the collision normal.
In order to know this you will be using the results of your plane tests. When you do collision testing against your polygon you can compute the distance of the point to each face plane. Keep track of the greatest signed value. If the greatest signed distance is negative then you know the point lies within the polygon. This greatest signed distance (which is still negative) comes from the plane which is the axis of minimum penetration. This distance is your penetration depth.
Once you have this axis you use the face normal of this plane as the direction of which to press these two objects apart. Exactly how you press them apart can be extremely simple or very complicated. Since you asked for something simple I'll recommend modifying position directly.
Take the position of your polygon and move it along the collision normal a distance equivalent to the penetration depth.
This resolution is called linear projection and it is very simple. You can also rotate your object accordingly, if you so wish. This gets more complicated. You can also apply an impulse to the velocity and integrate your velocity to move the position -this can result in a more stable and physically accurate resolve. If you want to know more about this try searching for "Impulse Resolution", I wrote some articles about this topic.