Imploringly is correct.
You are imploringly annoying!
"The sisters stared imploringly at their mother, each still holding a tiny stray kitten." "He stood by her school locker, looking up imploringly as he awaited her answer."
I asked my parents imploringly if I could go to the party, but they said I was too young.
to beg urgently or piteously, as for aid or mercy; beseech; entreat: They implored him to go.
beseechingly
Imploring means urgently begging. Two synonyms of imploring are soliciting and pleading. "The homeless man stood on the sidewalk, imploringly asking passers-by for change."
It's an adverb meaning "in a pleading manner".
Imploringly is the best I can think of. Perhaps people don't do enough things in a pleading manner to warrant a word.
The boy saw the cat up on the tree and felt solitously how will it come down.
The word gingerly is on page 17 of the UK edition. It is used in the line "patted Hagrid gingerly on the arm" when Professor McGonagall is comforting Rubeus Hagrid.
you mean what you mean