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There's this expression shown in Oxford:

have — going for one

Used to indicate how much someone has in their favour or to their advantage.

Why did she do it? She had so much going for her

In this expression, the pronoun coming after for always refers to the subject.

In the following 11 examples of the Oxford Dictionary, there's no example of a reflexive pronoun after for. Plus, the title of the expression has one instead of oneself.

So, I take it that you cannot have a reflexive pronoun come after for.

But then, there are numerous counterexamples of a reflexive pronoun coming after for in news articles and books:

...she was using blackorwhite thinking when she told herself that she had nothing going for herself... (book 1)

I'm not making any excuses for her but for a woman with so much going for herself Trevonne has low self esteem. (book 2)

One thing she had going for herself though was drive. (book 3)

Plus she had a lot going for herself. She was intelligent, funny, caring, ... (book 4)

Tonya makes it clear that skating was the only thing Harding had going for herself. (news article 1)

Even her attempt to have something going for herself outside the house, some kind of passion and hobby, got painted as tone deaf and self-indulgent... (news article 2)

So I wonder whether a reflexive pronoun in this expression is admissible.

My reaction upon reading every one of those counterexamples was, "That's obviously wrong!"

Have (something) going for you is and idiomatic expression from the early ‘60

If you have something going for you, or if something has something going for it, there are certain advantages that will make the end result successful:

• 
  They've got a happy marriage, great careers, wonderful kids - in fact they've got everything going for them.

(Cambridge Dictionary)

Have (something) going for one:

to have a talent, skill, etc., that helps one

• She's not as young as some of the other athletes, but experience helps, and she has that going for her. You should be more confident in yourself. You have a lot going for you!

(M-W)

Usage as evidenced by Google Books is most often correct with only a few instances of the use of the reflexive pronoun and all main dictionaries don’t use definitions or usage examples with reflexive pronouns.

I’d avoid using the reflexive pronoun even though the meaning is clear, so we could say that its usage is non standard.

While I certainly concur with the other answers on this subject, and the OP’s own feelings, that the use seems to be non-standard, I think what possibly is happening in the usage "have something going for oneself" is the the emphasis on one, from some aspect, by being the only recipient of the advantage, etc.:

As in the definition of oneself in ODO:

2 [emphatic] Used to emphasize that one does something individually or unaided.

‘the idea of publishing a book oneself’

Or in Macmillan:

2 used for emphasizing that you and not anyone else does something

It’s important to complete the application forms oneself.

If we consider this emphasis to be part of the usage, then all examples listed in the OP, as well as any other usage makes sense and the non-standardness is somewhat resolved.

This emphasis point seems to be supported also by the fact that many of the Google search hits point to celebrities or people who are somehow are considered outstanding and unique, just one example of the several:

As a solo artist this Toronto Reggae Singer has a lot going for herself. She writes, records, produces and is the Musical Ambassador for Read Across Jamaica Foundation

With all that said, lacking this intentional emphasis, I am neither able to find any definition or usage note that would justify using oneself instead of one, so in such uses it would be unquestionably non-standard.