What I'm calling a "crossum" is a puzzle much like a crossword, which uses digits rather than letters. The "clue" is the sum of digits in its solution. The digits 1 through 9 are used, and may not be repeated in a single answer.
Below is a solution chart with all possible digit combinations for each sum, arranged in columns of digit quantity.

For example, the only digits that can add up to 3 are 1 and 2.
The only digits that can add to 4 are 1 and 3 (2 and 2 violates the non-repeating rule).
The highest sum possible is 45, which is the sum of all nine digits.

Possible answers for each combination of digits and sum
2 3 4 5 6 7 8 9
3 1, 2
4 1, 3
5 1, 4
2, 3
6 1, 5 1, 2, 3
2, 4
7 1, 6 1, 2, 3
2, 5
3, 4
8 1, 7 1, 2, 4
2, 6 1, 3, 4
3, 5
9 1, 8 1, 2, 6
2, 7 1, 3, 5
3, 6 2, 3, 4
4, 5
10 1, 9 1, 2, 7 1, 2, 3, 4
2, 8 1, 3, 6
3, 7 1, 4, 5
4, 6 2, 3, 5
11 2, 9 1, 2, 8 1, 2, 3, 5
3, 8 1, 3, 7
4, 7 1, 4, 6
5, 6 2, 3, 6
2, 4, 5
12 3, 9 1, 2, 9 1, 2, 3, 6
4, 8 1, 3, 8 1, 2, 4, 5
5, 7 1, 4, 7
1, 5, 6
2, 3, 7
2, 4, 6
3, 4, 5
13 4, 9 1, 3, 9 1, 2, 3, 7
5, 8 1, 4, 8 1, 2, 4, 6
6, 7 1, 5, 7 1, 3, 4, 5
2, 3, 8
2, 4, 7
2, 5, 6
3, 4, 6
14 5, 9 1, 4, 9 1, 2, 3, 8
6, 8 1, 5, 8 1, 2, 4, 7
1, 6, 7 1, 2, 5, 6
2, 3, 9 1, 3, 4, 6
2, 4, 8 2, 3, 4, 5
2, 5, 7
3, 4, 7
3, 5, 6
15 6, 9 1, 5, 9 1, 2, 3, 9 1, 2, 3, 4, 5
7, 8 1, 6, 8 1, 2, 4, 8
2, 4, 9 1, 2, 5, 7
2, 5, 8 1, 3, 4, 7
2, 6, 7 1, 3, 5, 6
3, 4, 8 2, 3, 4, 6
3, 5, 7
4, 5, 6
16 7, 9 1, 6, 9 1, 2, 4, 9 1, 2, 3, 4, 6
2, 5, 9 1, 2, 5, 8
2, 6, 8 1, 2, 6, 7
3, 4, 9 1, 3, 4, 8
3, 5, 8 1, 3, 5, 7
3, 6, 7 1, 4, 5, 6
4, 5, 7 2, 3, 4, 7
1, 2, 6, 7
1, 3, 4, 8
1, 3, 5, 7
2, 3, 4, 7
2, 3, 5, 6
17 8, 9 1, 7, 9 1, 2, 5, 9 1, 2, 3, 4, 7
2, 6, 9 1, 2, 6, 8 1, 2, 3, 5, 6
2, 7, 8 1, 3, 4, 9
3, 5, 9 1, 3, 5, 8
3, 6, 8 1, 3, 6, 7
4, 5, 8 1, 4, 5, 7
4, 6, 7 2, 3, 4, 8
2, 3, 5, 7
2, 4, 5, 6
18 1, 8, 9 1, 2, 6, 9 1, 2, 3, 4, 8
2, 7, 9 1, 2, 7, 8 1, 2, 3, 5, 7
3, 6, 9 1, 3, 5, 9 1, 2, 4, 5, 6
3, 7, 8 1, 3, 6, 8
4, 5, 9 1, 4, 5, 8
4, 6, 8 1, 4, 6, 7
5, 6, 7 2, 3, 4, 9
2, 3, 5, 8
2, 4, 5, 7
3, 4, 5, 6
19 2, 8, 9 1, 2, 7, 9 1, 2, 3, 4, 9
3, 7, 9 1, 3, 6, 9 1, 2, 3, 5, 8
4, 6, 9 1, 3, 7, 8 1, 2, 4, 5, 7
4, 7, 8 1, 4, 5, 9 1, 3, 4, 5, 6
5, 6, 8 1, 4, 6, 8 1, 2, 3, 6, 7
1, 5, 6, 7
2, 3, 5, 9
2, 3, 6, 8
2, 4, 5, 8
2, 4, 6, 7
3, 4, 5, 7
20 3, 8, 9 1, 2, 8, 9 1, 2, 3, 5, 9
4, 7, 9 1, 3, 7, 9 1, 2, 3, 6, 8
5, 6, 9 1, 4, 6, 9 1, 2, 4, 5, 8
5, 7, 8 1, 5, 6, 8 1, 2, 4, 6, 7
2, 3, 6, 9 1, 3, 4, 5, 7
2, 4, 5, 9 2, 3, 4, 5, 6
2, 4, 6, 8
2, 5, 6, 7
3, 4, 5, 8
2, 5, 6, 7
3, 4, 5, 8
3, 4, 6, 7
21 4, 8, 9 1, 3, 8, 9 1, 2, 3, 6, 9 1, 2, 3, 4, 5, 6
5, 7, 9 1, 4, 7, 9 1, 2, 4, 5, 9
6, 7, 8 1, 5, 6, 9 1, 2, 4, 6, 8
1, 5, 7, 8 1, 2, 5, 6, 7
2, 3, 7, 9 1, 3, 4, 5, 8
2, 4, 6, 9 1, 3, 4, 6, 7
2, 4, 7, 8 2, 3, 4, 5, 7
2, 5, 6, 8
3, 4, 5, 9
3, 4, 6, 8
3, 5, 6, 7
22 5, 8, 9 1, 4, 8, 9 1, 2, 3, 7, 9 1, 2, 3, 4, 5, 7
6, 7, 9 1, 5, 7, 9 1, 2, 4, 6, 9
1, 6, 7, 8 1, 2, 4, 7, 8
2, 3, 8, 9 1, 2, 5, 6, 8
2, 4, 7, 9 1, 3, 4, 5, 9
2, 5, 6, 9 1, 3, 4, 6, 8
2, 5, 7, 8 1, 3, 5, 6, 7
3, 4, 6, 9 2, 3, 4, 5, 8
3, 4, 7, 8 2, 3, 4, 6, 7
3, 5, 6, 8
4, 5, 6, 7
23 6, 8, 9 1, 5, 8, 9 1, 2, 3, 8, 9 1, 2, 3, 4, 5, 8
1, 6, 7, 9 1, 2, 4, 7, 9 1, 2, 3, 4, 6, 7
2, 4, 8, 9 1, 2, 5, 6, 9
2, 5, 7, 9 1, 2, 5, 7, 8
2, 6, 7, 8 1, 3, 4, 6, 9
3, 4, 7, 9 1, 3, 4, 7, 8
3, 5, 6, 9 1, 3, 5, 6, 8
3, 5, 7, 8 1, 4, 5, 6, 7
4, 5, 6, 8 1, 4, 5, 6, 7
2, 3, 4, 5, 9
2, 3, 4, 6, 8
2, 3, 5, 6, 7
24 7, 8, 9 1, 6, 8, 9 1, 2, 4, 8, 9 1, 2, 3, 4, 5, 9
2, 5, 8, 9 1, 2, 5, 7, 9 1, 2, 3, 4, 6, 8
2, 6, 7, 9 1, 2, 6, 7, 8 1, 2, 3, 5, 6, 7
3, 4, 8, 9 1, 3, 4, 7, 9
3, 5, 7, 9 1, 3, 5, 6, 9
3, 6, 7, 8 1, 3, 5, 7, 8
4, 5, 6, 9 1, 4, 5, 6, 8
4, 5, 7, 8 2, 3, 4, 6, 9
2, 3, 4, 7, 8
2, 3, 5, 6, 8
2, 4, 5, 6, 7
25 1, 7, 8, 9 1, 2, 5, 8, 9 1, 2, 3, 4, 6, 9
2, 6, 8, 9 1, 2, 6, 7, 9 1, 2, 3, 4, 7, 8
3, 5, 8, 9 1, 3, 4, 8, 9 1, 2, 3, 5, 6, 8
3, 6, 7, 9 1, 3, 5, 7, 9 1, 2, 4, 5, 6, 7
4, 5, 7, 9 1, 3, 6, 7, 8
4, 6, 7, 8 1, 4, 5, 6, 9
1, 4, 5, 7, 8
2, 3, 4, 7, 9
2, 3, 5, 6, 9
2, 3, 5, 7, 8
2, 4, 5, 6, 8
3, 4, 5, 6, 7
26 2, 7, 8, 9 1, 2, 6, 8, 9 1, 2, 3, 4, 7, 9
3, 6, 8, 9 1, 3, 5, 8, 9 1, 2, 3, 5, 6, 9
4, 5, 8, 9 1, 3, 6, 7, 9 1, 2, 3, 5, 7, 8
4, 6, 7, 9 1, 4, 5, 7, 9 1, 2, 4, 5, 6, 8
5, 6, 7, 8 1, 4, 6, 7, 8 1, 3, 4, 5, 6, 7
2, 3, 4, 8, 9
2, 3, 5, 7, 9
2, 3, 6, 7, 8
2, 4, 5, 6, 9
2, 4, 5, 7, 8
3, 4, 5, 6, 8
27 3, 7, 8, 9 1, 2, 7, 8, 9 1, 2, 3, 4, 8, 9
4, 6, 8, 9 1, 3, 6, 8, 9 1, 2, 3, 5, 7, 9
5, 6, 7, 9 1, 4, 5, 8, 9 1, 2, 3, 6, 7, 8
1, 4, 6, 7, 9 1, 2, 4, 5, 6, 9
1, 5, 6, 7, 8 1, 2, 4, 5, 7, 8
2, 3, 5, 8, 9 1, 3, 4, 5, 6, 8
2, 3, 6, 7, 9 2, 3, 4, 5, 6, 7
2, 4, 5, 7, 9
2, 4, 6, 7, 8
3, 4, 5, 6, 9
3, 4, 5, 7, 8
28 4, 7, 8, 9 1, 3, 7, 8, 9 1, 2, 3, 5, 8, 9 1, 2, 3, 4, 5, 6, 7
5, 6, 8, 9 1, 4, 6, 8, 9 1, 2, 3, 6, 7, 9
1, 5, 6, 7, 9 1, 2, 4, 5, 7, 9
2, 3, 6, 8, 9 1, 2, 4, 6, 7, 8
2, 4, 5, 8, 9 1, 3, 4, 5, 6, 9
2, 4, 6, 7, 9 1, 3, 4, 5, 7, 8
2, 5, 6, 7, 8 2, 3, 4, 5, 6, 8
3, 4, 5, 7, 9
3, 4, 6, 7, 8
29 5, 7, 8, 9 1, 4, 7, 8, 9 1, 2, 3, 6, 8, 9 1, 2, 3, 4, 5, 6, 8
1, 5, 6, 8, 9 1, 2, 4, 5, 8, 9
2, 3, 7, 8, 9 1, 2, 4, 6, 7, 9
2, 4, 6, 8, 9 1, 2, 5, 6, 7, 8
2, 5, 6, 7, 9 1, 3, 4, 5, 7, 9
3, 4, 5, 8, 9 1, 3, 4, 6, 7, 8
3, 4, 6, 7, 9 2, 3, 4, 5, 6, 9
3, 5, 6, 7, 8 2, 3, 4, 5, 7, 8
30 6, 7, 8, 9 1, 5, 7, 8, 9 1, 2, 3, 7, 8, 9 1, 2, 3, 4, 5, 7, 8
2, 4, 7, 8, 9 1, 2, 4, 6, 8, 9 1, 2, 3, 4, 5, 6, 9
2, 5, 6, 8, 9 1, 2, 5, 6, 7, 9
3, 4, 6, 8, 9 1, 3, 4, 5, 8, 9
3, 5, 6, 7, 9 1, 3, 4, 6, 7, 9
4, 5, 6, 7, 8 1, 3, 5, 6, 7, 8
2, 3, 4, 5, 7, 9
2, 3, 4, 6, 7, 8
31 1, 6, 7, 8, 9 1, 2, 4, 7, 8, 9 1, 2, 3, 4, 5, 7, 9
2, 5, 7, 8, 9 1, 2, 5, 6, 8, 9 1, 2, 3, 4, 6, 7, 8
3, 5, 6, 8, 9 1, 3, 5, 6, 7, 9
4, 5, 6, 7, 9 1, 4, 5, 6, 7, 8
2, 3, 4, 5, 8, 9
2, 3, 4, 6, 7, 9
2, 3, 5, 6, 7, 8
32 2, 6, 7, 8, 9 1, 2, 5, 7, 8, 9 1, 2, 3, 4, 5, 8, 9
3, 5, 7, 8, 9 1, 3, 4, 7, 8, 9 1, 2, 3, 4, 6, 7, 9
4, 5, 6, 8, 9 1, 3, 5, 6, 8, 9 1, 2, 3, 5, 6, 7, 8
1, 4, 5, 6, 7, 9
2, 3, 4, 6, 8, 9
2, 3, 5, 6, 7, 9
2, 4, 5, 6, 7, 8
33 3, 6, 7, 8, 9 1, 2, 6, 7, 8, 9 1, 2, 3, 4, 6, 8, 9
4, 5, 7, 8, 9 1, 3, 5, 7, 8, 9 1, 2, 3, 5, 6, 7, 9
1, 4, 5, 6, 8, 9 1, 2, 4, 5, 6, 7, 8
2, 3, 4, 7, 8, 9
2, 3, 5, 6, 8, 9
2, 4, 5, 6, 7, 9
3, 4, 6, 7, 8
34 4, 6, 7, 8, 9 1, 3, 6, 7, 8, 9 1, 2, 3, 4, 7, 8, 9
1, 4, 5, 7, 8, 9 1, 2, 3, 5, 6, 8, 9
2, 3, 5, 7, 8, 9 1, 2, 4, 5, 6, 7, 9
2, 4, 5, 6, 8, 9
3, 4, 5, 6, 7, 9
35 5, 6, 7, 8, 9 1, 4, 6, 7, 8, 9 1, 2, 3, 5, 7, 8, 9
2, 3, 6, 7, 8, 9 1, 2, 4, 5, 6, 8, 9
2, 4, 5, 7, 8, 9 1, 3, 4, 5, 6, 7, 9
3, 4, 5, 6, 8, 9 2, 3, 4, 5, 6, 7, 8
36 1, 5, 6, 7, 8, 9 1, 2, 3, 6, 7, 8, 9 1, 2, 3, 4, 5, 6, 7, 8
2, 4, 6, 7, 8, 9 1, 2, 4, 5, 7, 8, 9
3, 4, 5, 7, 8, 9 1, 3, 4, 5, 6, 8, 9
2, 3, 4, 5, 6, 8, 9
38 3, 5, 6, 7, 8, 9 1, 2, 5, 6, 7, 8, 9 1, 2, 3, 4, 5, 6, 8, 9
1, 3, 4, 6, 7, 8, 9
2, 3, 4, 5, 7, 8, 9
39 4, 5, 6, 7, 8, 9 1, 3, 5, 6, 7, 8, 9 1, 2, 3, 4, 5, 7, 8, 9
2, 3, 4, 6, 7, 8, 9
40 1, 4, 5, 6, 7, 8, 9 1, 2, 3, 4, 6, 7, 8, 9
2, 3, 5, 6, 7, 8, 9
41 2, 4, 5, 6, 7, 8, 9 1, 2, 3, 5, 6, 7, 8, 9
42 3, 4, 5, 6, 7, 8, 9 1, 2, 4, 5, 6, 7, 8, 9
43 1, 3, 4, 5, 6, 7, 8, 9
44 2, 3, 4, 5, 6, 7, 8, 9
45 1, 2, 3, 4, 5, 6, 7, 8, 9
Yes, I did have fun doing this.