Here is the initial puzzle. As a reminder, the goal is to put letters into all the empty cells. R, A, and P are already included, so one must pick two more letters out of the nine given possibilities. The yellow highlighted cells are the locations of two four-letter words. All of the cells need to obey sudoku rules, i.e. five distinct letters in each row, column, and sector.
The rows and columns have natural names, counting from the top and the left, the first, the second, etc. The sectors don't have such natural ordering, so in this diagram I give the sectors numbers. This numbering is arbitrary, just so I can write about the solution steps.
Of course one can use whatever steps one likes to solve the puzzle - if the solution obeys the rules, then it is valid! And different instances will call for different approaches. But generally what seems to work well is to start off with as many sudoku-based steps as possible.
These puzzles have irregularly shaped sectors, so some steps are not too much like actual sudoku, but the reasoning is the same. The bottom row must have an R. But sector 4 already has an R, and almost all of the bottom row is in sector 4. The one cell in the 5th column is the only place in the bottom row where an R can be placed, so that's where it goes.
Column 5 needs a P, but two of the cells in that column are already filled in, and two are in sector 2, so there is only one place where a P can be placed in column 5, so that's where it goes.
Now sector 2 and column 4 need an A, which at first glance could be placed in row 1 or row 3. But look at sector 3. It also needs an A, but the only places an A can be placed are in row 3. So sector 2 cannot have an A in row 3! Therefore the A in sector 2 must be in row 1.
Sector 2 needs an R, but column 5 already has an R, so the R must go in the remaining cell in column 4.
Row 4 needs a P, but most of the empty cells are in sector 5, which already has a P. So the P must go in the one cell that is in sector 3.
Sector 3 needs an R, but most of sector 3 is in row 3, which already has an R. So the R must go in row 2.
Sector 1 needs an R, but columns 1 and 2 and row 2 already have Rs. So row 1, column 3 is the only possible place, and that's where it must go.
Sector 1 also needs a P. Row 2 and column 2 both already have Ps, so the only possible place left is row 1, column 1.
Column 3 needs a P, but there are already Ps in rows 2, 3, and 4. So the P must go in row 5.
At this point the sudoku has been pushed as far as possible. The sudoku rules, of course, don't give us any clue about how to pick the fourth and fifth letters of the puzzle. We have placed the given letters, A, R, and P in almost all the places they can go. There are two more As to be placed, in rows 2 and 3, and columns 1 and 3, but sudoku logic is not sufficient to determine how to place these.
Once we have exhausted the sudoku logic - which sometimes happens very early or sometimes, as with this puzzle, rather late - we need to think about what words might fit. Here we have more clues in the horizontal word: the last two letters are "RP". With the other available letters, the only word that comes to mind for me is "CARP".
Once we have decided one of the words, that should open up some further possible sudoku steps, which may provide more constraints on the other word.
Given "CA*P", and the R in row 1 blocking another R in row 4, the only word I can see that might fit is "CAMP". This determines the fifth letter, and the rest of the puzzle can be filled in very easily to complete the solution.
With some puzzle instances, the preliminary sudoku steps don't make either of the words so obvious. One may have to consider several different word possibilities, and look at the sudoku consequences of each possibility. The instances have been designed so that only one pair of words allows a complete sudoku solution, but just how the alternate possibilities will run into contradictions... sometimes it can be a bit subtle!
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