Question

A white face with edge pieces properly oriented but corner pieces mis-oriented by one position:

How can I shift all 4 corner pieces of the white face by one position?

Answer 1

You could just swap them by
- rotating the middle column (the one with 2 vertical reds) from back to front
- the middle piece is now at the bottom row with the red part on the hidden side of the cube
- Rotate the bottom row to left
- move the middle middle column back up again. Should have lost one of the white ones on the top surface
- now you have a white piece somwhere within the bottom row on the left. Look for it and rotate the bottom row until the single white piece is below its destined position (should be twice to right)
Next you can use the same move the other way round to unite the white piece with the top surface again
- move the bottom row to the left, lower the middle column(the one with 2 vertical greens) from back to front
- and let the previously moved white piece enter the white column from the left
- move the column back up again

This way you would automatically have the next white piece in the bottom row on the right and you could repeat from (2).

Answer 2

This is the first layer - just take them out and put them back in the right place!

If I built a single face first (which I don't) I would take all $4$ onto the bottom, turn the top a quarter then take them back onto the top like so: M2 S2 U M2 S2

(note: M is the slice between left and right, S is the slice between F and B - so M2 takes the white-red, white-orange and centre white onto the bottom, then S2 takes the white-green and white-blue onto the bottom while taking the centre white back to the top; U cycles the four corners, then M2 S2 undoes the moves to the bottom)

A complete list of the shortest, by face turn, algorithms ($9$ face turns):

R2 F  B' U2 L  U2 F' B  R2
R2 F' B  D2 R  D2 F  B' R2
R2 L2 F2 B2 U  R2 L2 F2 B2  <- this is the one I said I'd do
R2 L2 F2 B2 U  F2 B2 R2 L2  <- this one is very similar
F2 R  L' D2 F  D2 R' L  F2
F2 R' L  U2 B  U2 R  L' F2
F2 B2 R2 L2 U  R2 L2 F2 B2  <-   as are
F2 B2 R2 L2 U  F2 B2 R2 L2  <- these two!
L2 F  B' D2 L  D2 F' B  L2
L2 F' B  U2 R  U2 F  B' L2
B2 R  L' U2 F  U2 R' L  B2
B2 R' L  D2 B  D2 R  L' B2