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"A failure in this regard will further weaken what little political authority he has"

This is one from the editorial printed in an english newspaper here in korea.

But I think that 'what' in those purple letters is not proper.

I think it should be like just

"A failure in this reagard will further weaken little political authority he has"

or

"A failure in this regard will further weaken little political authority that he has'

Am I right? or, is that sentence above correct? and it it's correct sentence, plz let me know why 'what' can be used in that phrase.
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It is correct as written, or it could be "... will further weaken the little polical authority..."

It will weaken something... what will it weaken? It weakens the authority he still has. You can substitute "what" for the "the" in this type of construction:

"She took what little money she had and donated it all to the animal shelter."

"She gathered what little dignity she had left and left the room."

It does seem to work best with "what little" or when you are saying that there is not much of something.
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In this sentence, it's true that the politician's power/authority will be weakened, but it's not necessarily true that he's a "lame duck." There's no evidence from the initial sentence to indicate his status as a "lame duck."

A "lame duck" refers to a politician who is in his/her last term of office and can't run again for that office. For example, George W. Bush is serving his second four-year term as U.S. President. The U.S. Constitution bars him from running for a third term. Therefore, he's a "lame duck."
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"A failure in this regard will further weaken what little political authority he has"

He is a lame duck in politics, and some failure will weaken his power even further.

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