For example, in the sentence "The social worker implored the child to express his feelings about his new home", does the word "implore" make it seem like the social worker is more or less on his/her hands and knees begging the child, almost in a gaudy way? Or could it simply be interpreted as the social worker asking, but with passion in his/her job, for feedback?
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I did not ever see implore being used in the sense of getting on one's knees. I googled out of interest (after seeing your question) and did not find any dictionary offering that kind of a definition. All definitions of implore that I saw were in the lines of (from Cambridge dictionary):
to ask someone to do or not do something in a very sincere, emotional, and determined way: She implored her parents not to send her away to school.
Alternative meaning (again from Cambridge dictionary) for literary context (not in conversational use):
to ask for something in a sincere and emotional way: She clasped her hands, and glancing upward, seemed to implore divine assistance.
In the accompanying example, you can see that action of clasping one's hands (analogous to getting on one's knees) is explicitly mentioned alongside implore.