anrühren
to mix
To mix a powder, paste, or concentrate with a liquid in order to produce a usable mixture.
Ich muss noch schnell den Zement für die Reparatur anrühren.
I still have to quickly mix the cement for the repair.

to mix / to touch
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to mix
To mix a powder, paste, or concentrate with a liquid in order to produce a usable mixture.
Ich muss noch schnell den Zement für die Reparatur anrühren.
I still have to quickly mix the cement for the repair.

'Anrühren' (to mix/to move/to touch) is a separable verb. In a main clause, the prefix 'an-' moves to the end of the sentence. In subordinate clauses or in the infinitive with 'zu', the verb stays together.
Er rührt den Beton mit Wasser an.
He mixes the concrete with water.
The three meanings can be recognized well from the object. If it is about powder and liquid, 'mischen' (to mix) is meant. If the object is a person and the context is emotional, it is about feelings. If it is a prohibition, 'berühren' (to touch) is meant.
Für den Kuchen rühre ich das Puddingpulver mit Milch an.
For the cake, I mix the pudding powder with milk.
In the kitchen or in DIY work, 'anrühren' (to mix) is often used with powders that are mixed with a liquid. Typical examples are: 'eine Soße anrühren' (to mix a sauce), 'einen Teig anrühren' (to mix a batter/dough), 'Gips anrühren' (to mix plaster), or 'Zement anrühren' (to mix cement).
Kannst du bitte schon mal das Salatdressing anrühren?
Could you please go ahead and mix the salad dressing?
The emotional meaning of 'anrühren' (to move/touch emotionally) belongs more to elevated or literary language. It sounds more poetic and formal.
Das Gedicht war so schön, es hat mich sehr angerührt.
The poem was so beautiful; it moved me deeply.
In the meaning 'physically touch', 'anrühren' (to touch) is very often used in prohibitions and warnings in the imperative or with 'dürfen nicht' (must not / may not).
Warnung! Elektrischer Zaun. Nicht anrühren!
Warning! Electric fence. Do not touch!
to move
'bewegen' and 'anrühren' can both mean that something triggers feelings in a person. 'bewegen' is more general: a message, a film, or a speech can strongly affect or touch someone inwardly. 'anrühren' sounds more delicate and often a bit more personal. It fits especially with pity, sadness, or being emotionally moved. The words can be confused because both refer to an emotional effect. If the effect is generally strong, 'bewegen' fits. If something gently touches the heart, 'anrühren' sounds more natural.

Der Brief der alten Nachbarin rührte Frau Wolf tief an.
The letter from the elderly neighbor deeply moved Mrs. Wolf.
Der Bericht über die Familie hat mich sehr bewegt.
The report about the family moved me deeply.