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The speaker is experiencing unusual levels of politeness after visiting the house of a relative who wants them to do a favour (he didn't visit with the knowledge that she wants a favour):
「来たとたん何か用かよ」
As soon as I came can I help you?
I'm not sure which part I'm failing to understand. I was sure とたん meant 'the moment that'/'as soon as'. And I thought that 何か用か meant 'can I help you?', but when I put the two together it makes no sense at all.
While "Can I help you?" is one good way of translating 「何{なに}か用{よう}かよ？」 in some situations, this is not one of those situations.
Here, something like "So, you want a favor, eh?" would fit much better.
"As soon as I arrive(d), you want a favor, eh?"
I should perhaps mention the fact that "You want a favor (or something), eh?" is a far more literal TL of 「何か用かよ？」 than "Can I help you?" is. The latter is a fairly free translation.
That is because the 「よ」 part of the 「かよ」 ending makes the phrase feel/sound more like a statement than a question. 「かよ」 adds a mildly exclamatory or even accusatory tone to the sentence.
The speaker is saying,
"I just got here and you're already asking me for a favor?"
The use of かよ is a good hint that this is a bit aggressive/sarcastic, definitely not polite. Insert "huh" at the end of the sentence to get even more of the flavor.
This is the kind of thing you'd say if it looks like the person is using you for a favor rather than inviting you over because they want to hang out, etc.
The part is "-かよ". It is generally an expression of mild to moderate astonishment (that can be positive, neutral or negative) - a rough equivalent in English would be something like "Aha, so...?", or by extension "Is it really ok...?" when it is clear from the context that the speaker is "rightfully" expressing his/her astonishment (like "学校サボっていいのかよ" = "Is it really okay to skip school (today)?")
The appropriate translation for the original question has already been mentioned by others, but I have to disagree with @l'électeur as I believe that under no circumstances would "Can I help you?" be an appropriate translation for "何か用かよ". If it is to be interpreted as an offer of help, it would have been said light-heartedly to a friend, and much more informal, much less polite.
In fact, it is so informal/impolite that even the "formal" version (and when actually asked, unlike in "-かよ"), "何かご用でしょうか", is at risk of coming off as being passive-aggressive (cf. "Do you want something?"). If I am to, say, offer help to a stranger, I would much prefer something like "お呼びでしょうか？" ("Do you need something?")
Literal meaning of 用 in this context is "business; task". So 何か用(です)か means "Do you have some business (to discuss/for me)?" and "Can I help you?" is just a more polite version of that (although the actual polite version is more likely to be 何のご用ですか in Japanese). As @l'électeur says, a more direct variation will probably work much better here. I would probably use something like
I only just stepped inside, and you already want something [from me]?
(though I guess it does not differ that much)
