The structure of Bunsen flame tip under the influence of dilute, monodisperse inert (water) or fuel (methanol) sprays is theoretically studied using large activation energy asymptotics. A completely prevaporized mode is identified, in which no liquid droplets exist downstream of the flame. Parameters for open and closed flame tips in the analysis consist of the amount of liquid loading indicating the internal heat loss for the water spray or the internal heat loss and heat gain for the rich and lean methanol-sprays, respectively, and the (negative) stretch coupled with Lewis number (Le) which strengthens the burning intensity of the Le > 1 flame but weakens that of the Le < 1 flame, respectively. For rich methane-air flames (Le > 1) with water sprays (or lean methanol-spray flames with Le > 1), closed-tip solutions are obtained. The burning intensity of the flame tip is enhanced with either decreasing liquid-water loading (or increasing liquid-fuel loading) or increasing stretch. Conversely, the negative stretch weakens the burning intensity of a lean methane-air flame (Le < 1) with water sprays (or a rich methanol-spray flame with Le < 1) and eventually leads to tip opening, i.e., flame extinction. The burning intensity is further reduced with either increasing liquid-water (or liquid-fuel) loading or increasing stretch. Moreover, the open flame tip is further widened when either the liquid-water loading (or liquid-fuel loading) or the upstream flow velocity is increased. It is noteworthy that the gradual increase of liquid-fuel loading. strengthens the burning intensity of the lean methanol-spray flame (Le > 1) and thus leads to the transition of flame configurations from conventional Bunsen cone through planar flame to inverted flame cone (a convex flame shape with respect to the upstream reactants). The critical value of liquid-fuel loading, at which there exists a planar flame rather than a Bunsen cone flame, is increased with either increasing upstream flow velocity or decreasing equivalence ratio.