MO will have the primary right to tax the wages sourced to MO but paid to your client during the KS residency period.  Since this is a multi-state return, PTO wouldn't be able to figure out which income is sourced to and double-taxed to which state and how much tax was paid on it.  This will call for you to run some manual computations and enter at least the following under State & Local > Other State Tax Credit > KS Other State Tax Credit:

1. State abbreviation: This will be MO in your case;
2. Income tax liability in other state [Override]: This is the total liability on the MO return, which PTO will prorate for the amount assessed on the income subject to double taxation by KS and MO, where applicable;
3. Other state's adjusted source income [Override]: This is the MO equivalence of KS AGI on the MO return, which PTO will use, in part, to compute the effective tax rate on the MO return and, thereby, the limitations for other state tax credit based on MO tax assessed on the income subject to double taxation by KS and MO; and
4. Income earned in other state while Kansas resident (PY only): This is the amount of wages earned in MO during KS residency in your case.  PTO will use this piece of data to compute the KS tax assessed on this income and limit the credit to the lower of this or MO tax assessed on the same income.

Once that's done, double check the computation on the statement of K-40 to make sure the numbers make sense.