"If you would have let me, I would have came to New York."
There are two things going on here.
1. In Colloquial speech, Americans (and Canadians?) often tend to use 'would have' in the 'if'-clause instead of Standard 'had'. I don't know whether this is thanks to German, or whether it's an everyday variant that's climbing the social ladder to linguistic respectability, but American TV commentators of all stripes commonly use it: 'If he woulda just held on to that ball, it'da been six points'.
2. There's a long-standing tendency for the affirmative past tense form and the past participle form of irregular verbs to get mixed up in some speakers' speech. Given that these forms are often the same and that those of regular verbs are always the same, this can hardly be surprising, but it carries - or, at least, carried - a social stigma much like that of 'ain't': educated middle-class types will not succumb to this error. So, among certain sectors of society, one may hear non-Standard 'He coulda did it', 'She musta ate it', or, horrors of horrors, perhaps even 'They aint stole it' (personally, I've never heard the last one), but in the "right" sectors they were thrashed out of us as children - ironically, 'Speak proper!' is itself hardly grammatical, but no matter). I assume that 'came' is an instance of this. It's been around for centuries, but that someone should be touting it as correct (Standard?) is the bizarre thing.
Consequently, Standard English must read:
"If you had let me go, I would have come to New York."