Hcfyk

There are some positions in which you need to know how many moves have been played without pawn move or capture. For example:

[FEN "1R5K/8/8/8/8/8/7R/k7 w - - 98 1000"]

(1R5K/8/8/8/8/8/7R/k7 w - - 98 1000)

Here, white must play Ra2 or Rb1 to force a capture within 1 move. If this is not done the game is drawn due to the 50 move rule.

If the 50 move rule was not a concern it would be an easy mate in 2.

Of course it is very hard to remember how many moves have passed. Is white allowed to look at the scoresheet and count the number of moves during the game?

The rules for this and how it works are spelled out in articles 9.3, 9.4 and 9.5 of the FIDE Laws of Chess. They don't mention whether or not you are allowed to check your scoresheet to do this because you are always allowed to check your scoresheet. You would be very foolish to not check your scoresheet first because there are penalties for an incorrect claim.

Here are articles 93, 9.4 and 9.5 of the FIDE Laws of Chess:

9.3 The game is drawn, upon a correct claim by a player having the move, if:

9.3.1 he writes his move, which cannot be changed, on his scoresheet and declares to the arbiter his intention to make this move which will
result in the last 50 moves by each player having been made without
the movement of any pawn and without any capture, or

9.3.2 the last 50 moves by each player have been completed without the movement of any pawn and without any capture.

9.4 If the player touches a piece as in Article 4.3, he loses the right to claim a draw under Article 9.2 or 9.3 on that move.

9.5.1 If a player claims a draw under Article 9.2 or 9.3, he or the arbiter shall stop the chessclock (see Article 6.12.1 or 6.12.2). He
is not allowed to withdraw his claim.

9.5.2 If the claim is found to be correct, the game is immediately drawn.

9.5.3 If the claim is found to be incorrect, the arbiter shall add two minutes to the opponent’s remaining thinking time. Then the game shall
continue. If the claim was based on an intended move, this move must
be made in accordance with Articles 3 and 4.

Yes you're absolutely allowed to look at the score-sheet.

What you're not allowed to do is make special symbols/notes on the scoresheet to remind you of things. But that's non-applicable here.

Of course it is very hard to remember how many moves have passed. Is white allowed to look at the scoresheet and count the number of moves during the game?

In a serious tournament game, you would stop the clocks and ask the arbiter to come and check your scoresheets. I remember there was a recent incident between Russian GM Alexandra Kosteniuk against Chinese GM Ju Wenjun.

If there was no arbiter, you would have to do it with your opponents after stopping the clocks.