Position information:

Pip-count: Red 104-129 White
Score:       Red     0-0     White
Match:                   5-pointer

Red to play 5-4

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First make your choice and then read on.

Thanks for the rollouts. 

The real question is, "Why hasn't Red doubled?" This is a clear pass now and if Red hits, will always be a pass next turn. Even if not hitting looks minutely better when the game is played to a finish, hitting is correct. Why? Because if Red eschews the hit, White can take next turn after a 6-6. This is called Joker Control!

Quite right! I need new glasses! However I would still hit. Why? Because if you don't, White's play is trivially easy; build a board, wait for a shot or run if the race gets close. Hitting him makes for a much more complex game.

I can't match your BG skills even in my dreams, but I think that hitting makes the game much more complex for both sides, not just for white. As said in my blog, red has a relatively inflexible position, with 6 men stacked on the 6-pt and he might well leave a shot later. Thus by not hitting, he can simply get his men jump over white's anchor into safety without the fear of leaving a shot.

And will white really gain an edge in race after a 6-6?

Regards

Hitting maintains contact of course, but Red's play isn't very difficult from here if he hits is it?  Besides, you have to look for more contact and more gammons if you want a chance to outplay your opponent.
6-6 doesn't turn it round for White, but you surely would rather he rolls it from the bar! A play that makes 3 of his best rolls (6-6, 5-5, 6-5) into his three worst rolls is usually pretty good.
 For what it's worth, Snowie 3-ply chooses to hit, although I agree that a rollout marginally prefers not hitting. A very considerable factor in this decision is, how well will White use the 4 cube? A lot of players wait too long with a recube, particularly at this score, where Red has to pass 4 cubes that would be trivial money takes.

It's absolutely clear to run with both.
Staying achieves nothing really. White won't get a shot for ages with all those spares that Red has, so he must go and take a chance in the race, even though he trails by 10 pips after the roll. As almost all his wins will come when he rolls well and catches up, it is essential to free all his checkers for the race. You don't want good rolls like 4-4 and 3-3 to be blocked and you certainly don't want to be forced to run with one later, leaving a target behind.