First lets start with some definitions:
Frantically-in a distraught way owing to fear, anxiety, or other emotion.
Tentatively-in a way that lacks confidence; hesitantly.
Impromptu-done without being planned, organized, or rehearsed.
Usually when you need to handle babies very gently, so in this case frantically wouldn't work. Panic is another word for frantic, and when you are panicking gentle is not usually the first thing on your mind.
Impromptu is when you do something in the moment and don't really think much about it before hand. This wouldn't fit, because it really has nothing to do with a small child.
Tentatively on the other hand is the exact opposite. In many types of sentences it has a connotation or another meaning.<em> (Similar to saying "cheap" and "inexpensive". Even though they mean the same thing by definition, the connotation is different.) </em>Tentatively<em> </em>can be another way of saying gently so this answer would fit.
Tentatively
