Is the Gap Between the Window Header and the Header a Problem

it depends on a lot of things:

1) dead load above 2) live load (do you get lots of snow, rain, etc.) 3) wall finish (for practical reasons you may have difficulty changing the header on a stucco wall, which is what it looks like from the cement stains on the interior of the sheathing boards) 4) material. steel is much stronger than wood. spruce is stronger than pine.

in a 2 x 4 wall, you can do a double header construction with a 5/8" ply liner between them as a spacer. given this, the fact that you are increasing the span from 20" to 36", then the span will be almost twice what was there before, but its still only 36". generally, you need a 2x6 double header minimum for just about any scenario, so if you are worried about it, go to a double 2x8. here in ontario, lintels under 2x8 are not permitted (table a-12), but your span is so small it doesn't even fit the minimum requirements. just use a doubled 2x8. its overkill, but thats the best way to build anything.

If you fastened everything up already, be prepared to undo it. At least there is a presumption I am making that the old framing gave you the room you needed.

You may not need to undo everything, but I do see the common (outside) stud gone. The first thing is and this is a guess (that is, after you removed the damaged framing, you set the jack stud first, then the common stud last) on how you replaced the framing. The jack stud beside the window needs to be toe nailed into the bottom plate on the old marks left by the original jack. Making sure all the splinters, trash, old insulation is out of the way or will not move back behind the jack stud before it is set in place and secured. A lot of shaking goes on and things that are loose can go in places and keep things from going back in like they should. Nails from the old sheathing into the studs you removed are a good one for this. After all that is assured, finish nailing the jack to the header, insuring the face of the stud is flush with the original header. A little behind it is not a problem either, more room for drywall. Besides, the old framing you have may be 1 5/8" thick, instead of the 1 1/2" thick you have now.

As a footnote, check the wall with a straight edge to see which framing is out of line, creating the problem. Just a mention... it may help you understand a bit more of what is going on. If you used straight studs, good. If you used new framing that is bowed from the start, not good. Again, these things are to help you understand what you have, I can't do anything from here but make suggestions.

Once you have all the things done I mentioned above, and the jack stud is in place and the straight edge has confirmed everything to be flat and the jack stud should be showing hopefully the 1/2" you need for drywall, if so move on with adding the last stud and drywall.

If you are willing to undo even more nailing before you try the above fix, I see the header is slightly proud of the face of the stud beside it. If this means there is room behind the header this will help move the jack stud back when the header moves back also. I see perhaps an 1/8" of the 1/4" gone that you need. After thatr has worked hopefully, toenail the jack stud so it draws the jackstud back to expose the remaining window jamb you need and tighter still to the common stud.

This is also another presumption that the window, since so much has gone on around it that the window has not simply popped out of the rough opening a bit since the fastening has been violated by the replacing of the jack stud, meaning it only needs to be driven back in from the outside using a block of wood to protect the window from the direct hammer blows and refasten it in place so it doesn't pop out again.