This is the 14641st most frequent Russian word.
переложив
"Переложив" means "having shifted" or "having transferred."
The word 'переложив' here is used as a gerund meaning 'having transferred' and describes the completed action of moving the pie as part of finishing the recipe.
Я закончил рецепт, переложив пирог на тарелку.
I finished the recipe by transferring the pie onto a plate.
'Переложив' is used again as a gerund to indicate the action of moving the pen between hands that occurred concurrently with writing the letter.
Она писала письмо, переложив ручку из правой руки в левую.
She was writing a letter by shifting the pen from her right hand to her left.
In this sentence, 'переложив' indicates the act of transferring responsibility and serves to explain the cause of feeling relieved.