In a cottage that is unheated in the winter, snow sits on the eaves trough and when it melts it finds a way to seep into the cottage, run along the rafters and get leak into the living area. How can I best stop this seepage?
That 2x4 is probably your problem, if snow and ice build up in the gutter (aka eaves trough) high enough then water will flow into the spot where you can see debris there now.
The best thing you could do, without moving the eaves trough and cutting the roof back, is probably to flash with metal. First, I would cut a piece of wood at an angle to sit on top of that 2x4. Then, get a wide piece of flashing. Tuck it up right under the metal roof, and run it down the vertical section, down the newly installed angled piece, and into the back of the eaves trough. That way any water that gets to that space between the roof and the eaves trough can't get behind the flashing, and will eventually make it's way back down.