Combine the across and down clues to solve each square.Let us start with the last cell, where the down clue is 9 and across clue is 16.
The only two digits which can produce a sum of 16 is 9 and 7 ( 8 and 8 cannot be used).
The down clue is 9, which makes the highest digit possible in that number is 8. The last cell cannot be 9, It is 7.
You can easily find the adjacent cells to solve the across and down clues.
The across clue in fourth row is 19 for 3 cells. We already found the last cell to be 2. The two remaining cells must account for a sum of of 17, which means the digits 9 and 8 will occupy those cells. Since 9 is already used in fourth column, 9 goes to third column and 8 goes to the fourth column.
With this introduction, I hope you can solve the rest of the puzzle.