The pKa for a weak acid to be used in the buffer should be as close as possible to the desired pH. What if the pKa is not close to the desired pH? What is the problem with choosing such a weak acid used in the buffer?

When pH = pKa, the ionizable compound in question (either acid or base) will be half protonated and half deprotonated

Explanation:

A convenient way of expressing the relative strength of an acid is by the value of its pKa, which makes it easy to see in small changes in pKa the changes associated with large variations in Ka. Small pKa values are equivalent to large Ka (dissociation constant) values, and as pKa decreases, the strength of the acid increases.

An acid will be stronger the lower its pKa and in a base it happens the other way around, which is stronger the higher its pKa.

Those dissociation constants are not fixed, they depend on other variables. For example, the dissociation constant changes at different temperatures. However, it maintains its value at the same temperature, before changes in the concentration of any of the species or even under the action of a catalyst.