Full solution with explanation
Of the numbers from one to nine:
- ONE, TWO, and SIX have three letters; we already know which region is ONE, and TWO and SIX are then also given by the W and X already in place.
- FOUR, FIVE, and NINE have four letters; FOUR is given by the U already in place, NINE must be the bottom right region, so FIVE must be the top middle one.
- THREE, SEVEN, and EIGHT have five letters.
Since "the space you land on must be the first letter of the word", we can immediately fill in the rest of the TWO and the first N of NINE.
Now the F of FOUR is reachable in two moves from the T of TWO, so it must be either below the U or the lowest cell of the FOUR region. The latter is impossible since the intermediate step must be in one of the five-cell regions. This also tells us where the T of THREE must be, so THREE is the rightmost five-cell region:
Then we know exactly where the F of FIVE must be, and we can fill in the whole of SIX. Since the third row must contain at least one E, it must be the second E of THREE, which gives us the "one exception" to letter duplication in a single row/column. (Clearly this "one exception" means the entire letter E, rather than the single pair of E's we can already see adjacent in the initial setup.)
Now that no-duplication rule tells us that the O of FOUR can't be in the third row and the R of THREE can't be in the fifth column, enabling us to finish filling in those regions.
The E of EIGHT (the only E in the region) must be two moves away from the S of SIX. This tells us which way round the SEVEN and EIGHT regions are, and where the S of SEVEN must be.
Also, the no-duplication rule tells us the N of SEVEN can't be in the first column, and then the fourth (SEVEN-specific) rule enables us to complete this region.
The no-duplication rule also tells us where the T of EIGHT and the second N of NINE must be:
Now we have the I,V,E of FIVE, the I,G,H of EIGHT, and the I,E of NINE to fill in. The fourth and fifth columns must each contain an E, so we can fill in the two last E's. The no-duplication rule tells us where the I of EIGHT must be, and the rest is free. So the final, most general answer is: