Show us code. Hard to know what you're describing otherwise.

(I haven't needed "a let binding in each arity that was basically the same" so I'm not sure what you're trying to do)

(defn format-query
    ([a] (let [calculated-a (somefunction a)
               other-stuff-based-on-calculated-a (-> calculated-a
                                                     otherfunction
                                                     str)]
             (format query-defined-elsewhere calculated-a other-stuff-based-on-calculated-a "")))
    ([a b] (let [calculated-a (somefunction a)
                 other-stuff-based-on-calculated-a (-> calculated-a
                                                       otherfunction
                                                       str)]
               (format query-defined-elsewhere calculated-a other-stuff-based-on-calculated-a b))))

I basically format the query in the exact same way between the two but have the additional b parameter

You can have one arity call the other passing the missing param, i.e.

(defn format-query
  ([a] (format-query a ""))
  ([a b] (let [calculated-a (somefunction a)
               other-stuff-based-on-calculated-a (-> calculated-a
                                                     otherfunction
                                                     str)]
           (format query-defined-elsewhere calculated-a other-stuff-based-on-calculated-a b))))

of course

thank you