I'll analyse this situation for a special case.
If the hydrogen gas was already the limiting reactant, then increasing the amount of CO does not increase the theoretical yield, but it does push the equilibrium further to the right, so the percentage yield would increase.
What does the question mean by yield though? If it just means the amount of product formed, then a simple application of Le Chatelier's principle would suffice. If it means percentage yield, then you'll have to take into account situations when the reaction is an excess reaction or when either reagent is in excess. That does not look like fun.
And yes, the K value is constant if you add CO, but the denominator of the fraction increases, so the numerator must increase too.