Some general observations:
- Each 3x3 box contains exactly three mines. They can't be all in a single row, column, or diagonal, since then the $1$ in that box would be unplaceable. If the $3$ in a box is in the corner, then the $1$ must be in the opposite corner. The centre cell of a box can never be $1$ or $2$.
- Two other forbidden combinations of mine placement are $\{$top-left, top-middle, middle-left$\}$ and $\{$top-left, top-middle, bottom-left$\}$ (also rotations and reflections of these patterns), since then the $2$ in that box would be unplaceable.
- Every mine cell must be at least $4$, so in Killer Sudoku boxes or row/column sums, two mines can give a sum from $9$ to $17$ inclusive, three mines from $15$ to $24$ inclusive, four mines from $22$ to $30$, five mines from $30$ to $35$, six mines must be $39$. Also, if some mine cells sum to $9$, then they must be either a single cell containing $9$ or two cells containing $4,5$, while if some mine cells sum to $10$, then they must be two cells containing either $4,6$ or $5,5$.
Of particular interest in this puzzle:
Killer Sudoku Box 14 has two mines ($5,9$ or $6,8$). Column 28 has four mines ($4,7,8,9$ or $5,6,8,9$).
Step-by-step deductions
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Start with the Killer Sudoku box summing to
$34$.
It can contain at most five mine cells (three from the top-middle box, two from the top-right box), but less than five can't sum to $34$, so it must be exactly five. Therefore all the mines of the top-middle box must be in that Killer Sudoku box. Then there's only one possible position for the mine adjacent to the already-placed $1$, and the $3$ must also be in that box. Using red for mines and grey for not-mines, we have this, with the numbers inside that 2x2 square being $3$ and three of $9,8,7,6,4$ (these being the five mine cells summing to $34$).
Also, the Killer Sudoku box summing to $22$.
It can contain at most four mine cells (the lower three of the five can't all be mines), but less than three can't sum to $22$, so it must be three or four. If it's three, they must be $9,8,5$ or $9,7,6$; if it's four, they must be $4,5,6,7$.
And the Killer Sudoku box summing to $27$.
By the general observations above about what given numbers of mines can sum to, this box must contain exactly four mines, $4,7,8,9$ or $5,6,8,9$. In the bottom-left and bottom-middle boxes, that's either three (counting the centre cell) and one, or two and (both) two. Also the two cells in the $9$ diagonal can't both be mines, because if there are two mines in this diagonal they must be $4$ and $5$, which can't both appear as mines in the $27$ box.
Now consider the row with $32$ sum.
Any row or column can contain at most six mines. Here some of the numbers $4,5,6,7,8,9$ must sum to $32$, so it can only be the five numbers $4,5,6,8,9$. Considering the top three 3x3 boxes, two of them must have two mines in the middle row, and the third must have exactly one mine in the middle row.
This is still a very partial answer, but it's a very hard puzzle! Maybe someone else can continue from here, using some of my methodology and deductions, or I'll come back later to expand on this.