Is there a path to victory? I played a game against my mobile AI, me playing white. At this point in the game I had to get off my train, and when I picked it up again I was a bit quick on the trigger, moving my e-pawn, which cost me my rook and the game.

8/pk3rpp/2R5/8/P7/4P3/6PP/6K1 w - - 0 1


I've been scratching my head for half a day trying to figure out whether it was really a won game. My question here is, what is worth more: my pawn, or his king being more free to move around?

An additional pawn (and a passed one at that) is a strong piece in the end game, but so is a king, and my pawn is far from promotion. If I could somehow protect my a-pawn while simultaneously guide the e-pawn forwards I would probably win. Is that doable?
Yes, this is a winnable endgame for white. Though you have to work hard for it, precise thinking would lead you to definite win. 

It's hard to explain all moves that could lead to win or draw because then there would be lots of ifs and then, so I will try to explain you the plan where you can have the most possible opportunity for a forced win.

As you said king is a very important piece to be in play in endgame, and here, not only is your king cut off from the rook on f file but black's king is also cut off by your rook on 6th rank. To start with you can play 1.Re6 not allowing black's king to come forward beyond the 7th rank and now planning to march the e pawn. But there is no urgency to march it right away as the e pawn is completely safe on e3 as white can defend it by playing g3 first and put your king on g2 so if any time black plays Rook to e file attacking the pawn then white king can come to f3 and defend it. Also there is no good way Black can attack your pawn on a4. The Blacks King can not come forward and the rook on f7 is stuck because if he moves it down the board then white is going to take control of the 7th rank and cause problems for g and h pawns and black king is far away from defending it (g and h pawns can not be pushed up because of Rook coming behind those pawns and capturing it and if the black rook comes to defend it, then your e pawn is free to move up the board), if the rook moves sideways then white king can just come into play by playing Kf3. At some point you can even give up your e3 and a4 pawns for attacking black pawns on g and h files, the basic idea here is to allow black's 'a' pawn to march forward while getting the advantage of your own 2 connected passed pawns to march forward to queen, and here White wins as 2 pawns down the lane is going to be much more effective than Blacks a pawn.

I believe I have given you a rough plan to understand how you can win this endgame, if you have any idea of defence or any doubts, please comment and let me know. If you want to have precise moves and deep analysis of each move, I suggest you to put this position into a chess engine and see what happens when you play different moves, but again there, you will understand the basic idea and the plan to look forward to from the initial position, is the same as I have mentioned.