Adding a formula with relative cell address to data cells quickly

From what you are saying, you want to change to original number. IF Budget/July in B3 = 500, The manager inputs 50% for July, then the figure in B3 should change to 250.

You could setup the original budget template with the original budget figures.
On a separate sheet, copy the outlay of the budget (copy sheet) and replace the figures with = the formula x multiplayer which would be set as 100%. If you/manager then changes the percentage, the linked figures will change.

If so, you could use Past Special. You need to do it once for every range multiplied with the same figure. If you have lots of numbers in column B (July) and all need to be multiplied with 50% (F3), then do this:
Click F3 and copy (Ctrl+C).
Highlight all cells you want to multiply with the value in F3.
Right-click and select: Paste > Paste Special (Ctrl+Alt+V)
Select multiply
Click OK

An alternative to retain the original figures, or using as a helper sheet (Sheet2);
You could setup the original budget template with the original budget figures.
On a separate sheet, copy the outlay of the budget (copy sheet) and replace the figures with = the formula x multiplayer which would be set as 100%. If you/manager then changes the percentage, the linked figures will change.

I think the example doesn't give enough information but it could be as straightforward as B3*f$3 copy down and to the right