Factor + math.unicode, 35 bytes

[ [1,b] swap '[ _ ^ -1 ^ ] map Σ ] 

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Explanation:

It's a quotation (anonymous function) that takes two integers from the data stack (in m, n order) as input and leaves a fully-reduced mixed fraction (it's simply the way that Factor is) on the data stack as output.

• [1,b] swap Make a range from 1 to n and bring m to the top of the stack.
• '[ _ ^ -1 ^ ] Push a quotation to the stack for map to use later. '[ _ says to slot whatever is on top of the data stack (m) into the _.
• map Apply a quotation to each element of a sequence, mapping the results into a sequence of the same size.
• ^ Raise a sequence element to the m power.
• -1 ^ Take the reciprocal. This is shorter than recip and 1 swap /.
• Σ Sum.

Factor + math.unicode, 35 24 bytes

[ [1,b] 0 rot - v^n Σ ] 

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-11 thanks to @Bubbler!

Explanation:

It's a quotation (anonymous function) that takes two integers from the data stack as input and leaves a fully-reduced mixed fraction (it's simply the way that Factor is) on the data stack as output.

• [1,b] Make a range from 1 to n.
• 0 rot Push 0 and bring m to the top of the stack.
• - Subtract m from 0.
• v^n Raise every element in the range to the -m power.
• Σ Sum.