Shifting notes left-right within a staff

No need to delete the rest or have MuseScore try to guess how may notes to move. Just move the notes directly. Select the range of notes you want to move (eg, click first note, Shift+click last), Ctrl+X to cut, then click where you want to move the notes to, Ctrl+V to paste. The notes will overwrite the rest.

1. Select the first note of the passage you wish to move. (In your example, it looks like you would want to click on the first eighth note of beat 2 in the measure.)
2. Shift-click on the last note of the section you wish to move. (This will highlight the entire passage to be moved.)
3. When all the notes are selected (turn blue), cut and paste the selected passage onto the quarter rest of beat 1. (This pastes all the notes to start on beat 1 - essentially 'moving them over'.)

See:
https://musescore.org/en/handbook/copy-and-paste-0

Regards.