wickeln
to wrap
To wind something, e.g. a scarf, thread, or bandage, around something else, or to roll it up into a roll or ball.
Er wickelte sich einen warmen Schal um den Hals.
He wrapped a warm scarf around his neck.

to wrap / change
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to wrap
To wind something, e.g. a scarf, thread, or bandage, around something else, or to roll it up into a roll or ball.
Er wickelte sich einen warmen Schal um den Hals.
He wrapped a warm scarf around his neck.

You can recognize the meaning of “wickeln” (to wrap/change) from the object and the prepositions. If “um” (around) appears with it, it is usually about wrapping something around something. If the object is a baby, it means changing a diaper. The expression “um den Finger wickeln” (to wrap around one’s finger) is a fixed idiom.
Sie wickelte das Geschenk in buntes Papier.
She wrapped the gift in colorful paper.
In the meaning “to wind around something,” “wickeln” (to wrap) usually needs either two accusative objects or one accusative object and a prepositional phrase with “um” (around). The structure is: “etwas (Akk.) um etwas (Akk.) wickeln” (to wrap something around something).
Die Mutter wickelte eine Decke um das schlafende Kind.
The mother wrapped a blanket around the sleeping child.
When you say “Ich muss das Baby wickeln” (I have to change the baby), it always means changing the diaper. You do not need to add the word “Windel” (diaper); it is clear from the context.
Das Baby schreit, vielleicht muss man es wickeln.
The baby is crying; maybe it needs to be changed.
The expression “jemanden um den Finger wickeln” (to wrap someone around one’s finger) is a fixed idiom. It means that you influence a person with charm so that they do everything you want.
Die kleine Tochter wickelt ihren Vater mühelos um den Finger.
The little daughter effortlessly wraps her father around her finger.
to influence
'beeinflussen' and 'jemanden um den Finger wickeln' (to wrap someone around one’s finger) both have to do with one person having an effect on another. 'beeinflussen' is more neutral and can be used for opinions, decisions, or behavior. The effect can be open or indirect, and it does not have to be bad. 'jemanden um den Finger wickeln' is a fixed expression with 'wickeln'. It means that you easily get someone to do something through charm or clever words. This sounds more personal and often a bit critical or playful. 'beeinflussen' fits in many situations; 'um den Finger wickeln' only fits when someone acts on another person very skillfully.

Mit ihrem freundlichen Lächeln wickelt die kleine Tochter ihren Vater schnell um den Finger.
With her friendly smile, the little daughter quickly wraps her father around her finger.
Die Meinung seiner Freunde beeinflusst Max bei seiner Entscheidung.
The opinion of his friends influences Max in his decision.