Cryptic Division 9: Sent to the Clue Factory

Question

This is a word division puzzle which uses cryptic clues. If you're unfamiliar with either or both of those, you can click the associated link.

In order to solve the alphametic, you'll first need to fill in the dividend, divisor, and quotient by solving the cryptic clues. I've left the enumerations off to provide a bit of extra challenge. Once you fill those in, the puzzle should be solvable with only arithmetic and logic. The solution is a 10-letter word or phrase found by ordering the letters from 0 through 9. A complete answer should provide this solution along with explanations of the cryptic clues and your path through the alphametic.

As always, I've created an interactive version that will autofill from the grid to the clues and vice versa. Have fun!

Clues:

Decrease payments to accommodate daughter
Pouring from copper pot's side
A team of men coming first, winning English trophy

Except ... well, I was hoping I wouldn't have to mention this, but there's probably no way around it. There was a bit of an incident when I was making these clues, and the wordplay halves were split from the definition halves! I put them back together in the way that made the most sense to me, but something still seems wrong. I'm sure you can figure it out, though - you're very smart, and extremely cool, and you have great taste in puzzles.

Accessible version:

      ???
    -----
???|?????
    EHI
    ---
    WTHE
    WTSC
    ----
      HIS
      CUP
      ---
       CW

Answer 1

The clues:

As the text suggests, the definitions and wordplay have gotten mixed up among the three clues! The answers are:
- Decrease (def: CUT) payments to accommodate daughter (D inside dues = DUDES)
- Pouring (def: WET) from copper pot's side (Cu+ last letter of pot = CUT)
- A team of men (def: DUDES) coming first, winning English trophy (first letters give WET)

The division problem:

We know DUDES has to be the dividend, but which of CUT and WET is the divisor? From the last step of the division we see that CUP is a multiple of the divisor, so it must be WET. The second subtraction HE - SC = HI immediately tells us that S is 0:

Next step:

The final subtraction tells us that W + P = 10 and the three multiplications are:
WET * C = EHI
WET * U = WTSC
WET * T = CUP
Either C or T must be at least 3, so since the first and third multiplications both result in three-digit numbers that tells us that WET * 3 < 1000. Therefore WET * 9 < 3000, so the second multiplication tells us that WTSC <= WET * 9 < 3000 and therefore W is 1 or 2.

The third multiplication also tells us that T * T ends in the digit P, so P must be 1, 4, 5, 6, or 9. Together with W + P = 10 the only possibility is W = 1, P = 9:

Finally:

1ET * T = CU9, so T must be 3 or 7. If T is 7 then the only possibilites for this multiplication are 127 * 7 = 889 or 137 * 7 = 959, both of which have repeated digits so don't work. Therefore T = 3 and 1E3 * 3 = CU9.

A quick runthrough of all possible values for E finds that only 6 or 8 won't lead to clashing digits for E, C, and U; however, from the first subtraction we know E < D so E can't be 8 since it would force D to be 9 which is taken. Therefore E = 6, C = 4, U = 8:

The last few digits are now straightforward, giving us the final answer:

SWITCHED UP, which also describes the definitions and wordplay in the cryptic clues at the beginning!