The combined excitation of high density potassium (K) vapour by 100 fs pump-coupling pulses is experimentally studied. The intense pump pulse excites the two-photon $\left|4{\text{S}}_{1/2}\right.〉\leftrightarrow \left|6{\text{S}}_{1/2}\right.〉$ transition and internally generated emissions are initiated along the atomic paths: $\left|6{\text{S}}_{1/2}\right.〉\leftrightarrow \left|5{\text{P}}_{3/2,1/2}\right.〉\leftrightarrow \left|4{\text{S}}_{1/2}\right.〉$ (path-1) and, $\left|6{\text{S}}_{1/2}\right.〉\leftrightarrow \left|4{\text{P}}_{3/2,1/2}\right.〉\leftrightarrow \left|4{\text{S}}_{1/2}\right.〉$ (path-2). The temporally delayed coupling pulse coherently drives the $\left|6{\text{S}}_{1/2}\right.〉\leftrightarrow \left|4{\text{P}}_{3/2,1/2}\right.〉$ transitions, in a Λ-type excitation scheme. The competing axial and conical emission components of the well-resolved $\left|4{\text{P}}_{3/2,1/2}\right.〉\leftrightarrow \left|4{\text{S}}_{1/2}\right.〉$ transitions (D₂ and D₁ lines of K) are substantially enhanced and controlled, for appropriate detunings and pump-coupling temporal delays. The coherence relaxation time (CRT) of the two-photon excited $\left|6{\text{S}}_{1/2}\right.〉$ state is determined by exploiting the temporal delay in the pulse sequence. The effect of the pulse delay and the fs pulse bandwidth on the system dynamics is discussed as well as the role of dephasing collisions between K and buffer gas atoms. The proposed scheme can be employed in radiative multi-level systems, for the direct estimation of coherence relaxation rates of various states.