The letters correspond to the numbers 1 to 26 in the order AXOKRBUZDYHJVCTQNFGIEMLPSW. The digits in the completed grid, shown here in full, read in natural order, decode as SHADE EVENS BUT RUB OUT ALL OF ODDS: ERGO X.

The grid numbers shared by across and down entries (where one always has 3 digits while the other has 4) are 1, 5 and 13, which must correspond to A, R and V in some order.

Clue K means H has 2 digits and K is one of {4, 12, 17, 23}.
The R clue with HHH thus gives a 4-digit answer,
so the other R clue means OKKK has 3 digits, so **K = 4**.

Clue U’s answer is a bit less than UUUU; the only possible entries with the right number of digits are 7dn = 2111 and 9ac = 5917 (not 5ac = 529, because there’s only one U clue).

Since A and V are both odd, the answer to the second V clue is even, whether X is even or odd.
If V is 5, then the answer has a factor of 5 (V), so it ends in 0,
which isn’t allowed for either 5ac or 5dn.
If V is 1, the answer to L is either 7 or 9 (U), which doesn’t have enough digits for any entry.
Thus, **V = 13**.

From the first V clue (whose answer has 3 or 4 digits), O < 7.
It can’t be 1 or 5 (taken by A and R) or 4 (taken by K).
If O is 6, the first V clue gives 9059 (if U is 7) or 11649 (if U is 9),
neither of which fits at 13ac or 13dn.
Clue O is QU, which can have at most 3 digits, so it can’t be 2dn.
Thus, **O = 3**.

From clue W, XXXXX < WW, which can be at most 676, so **X = 2**
(not 1, because there’s only one X clue, and not 3 because it’s taken by O).

The second R clue now gives 192, which means the first R clue has a 4-digit answer starting with 1,
so H is 11 or 12 (not 10, since 4dn can’t end with 0).
The first R answer is thus either 1363 or 1760, but neither 1ac nor 5dn can end with 0, so **H = 11**.
This is entered at 4dn, so 9ac starts with 1, which rules out 9ac = 5917 (above), so **U = 7** (7dn = 2111).
Also, 1ac ends in 1, so it can’t be the first R answer, thus **R = 5** (5ac = 192, 5dn = 1363) and **A = 1**.

The two V clues now give 557 and 5512 for 13ac and 13dn respectively.

The second A clue is (2D + 13)DD, so to avoid having more than 4 digits,
D must be at most 15; the available values are {6, 8, 9, 10, 12, 14, 15}.
D can’t be 6 because that would make its clue, 16DD + S, give 576 + S for 6dn,
where we know the first digit is 9.
For the remainder, the answer to clue D has 4 digits, which rules out all but 9 and 14.
We know the first digit of 14dn is 7, so it can’t be 3136 + S, so D isn’t 14;
thus **D = 9**, which gives 1ac = 2511 for the second A clue.

The D answer is now 9ac = 1296 + S, which must match 13_1,
so S is 15 or 25 (not 5, which is taken by R).
The X clue (2dn = 5__5) is the first A clue (1dn = 2__) multiplied by S, so **S = 25** and 9ac = 1321.
The X clue now gives 25(BB + 21B + 49) = 5__5;
if B is 8, we get 7025, which is too high, so **B = 6**, 2dn = 5275 and 1dn = 211.

Y’s answer is now 17256, which must be either 10ac or 19ac,
so 11dn either begins with 5 or ends with 7.
Its clue (H) is WW − 113, so if it ends in 7, W must be 20,
but that would give 20dn = 1104, which doesn’t fit.
So **Y = 10** (10ac = 17256).
To match 11dn = 5__, **W = 26** (not 25, which is taken by S), 11dn = 563 and 26ac = 1932.

The M value is 1839, which now fits only at 22ac, so **M = 22**.
This makes 23dn = 91, which is the L value, so **L = 23**.
The P value is 925, which now fits only at 6dn or 24ac, but 6 is already taken, so **P = 24**.
The F value is 181, which now fits only at 18dn, so **F = 18**.
The T value is 371, which fits only at 15ac, so **T = 15**, making 6dn = 927.
The G value is now 23233, which fits only at 19ac, so **G = 19**.
That makes the C value 7235, which fits only at 14dn, so **C = 14**.

The only remaining place for the Z value of 121 is 8ac, so **Z = 8** and the grid is complete.
From the O clue (3dn), 112 = 7Q, so **Q = 16**.
From the Q clue (16dn), 7323 = II + 6923, so **I = 20**.
From the I clue (20dn), 299 = 320 − E, so **E = 21**.
The remaining unassigned entries are 12ac = 71 and 17ac = 51,
of which only the latter can be the N value of 3N, so **N = 17** and **J = 12**.

The grid digits, in row order, with the immediately forced letter breaks, are:

25, 1119, 212113, 2117, 25, 6, 7, 15, 5, 7, 6, 3, 7, 15, 123, 23, 3, 18, 3, 9, 9, 25, 215, 19, 3, 2.

Converting the forced digits to the corresponding letters, we get:

25, 1119, 212113, 2117, 25, B, U, 15, R, U, B, O, U, 15, 123, 23, O, 18, O, D, D, 25, 215, 19, O, X.
With a bit of trial and error, this leads to SHADE EVENS BUT RUB OUT ALL OF ODDS, ERGO X.