Generally two light beams crossing each other do not interact, regardless of whether the sources are coherent or incoherent. So even if the 2nd source were a laser (a coherent source), you would not see an interaction between the two beams.
If the 2nd source were a laser that was phase locked to the first source (meaning, the two sources are not just coherent, but mutually coherent), it would produce an interference pattern where the two beams crossed.
Otherwise, the main way you might produce an interaction between the two beams would be if they crossed in a nonlinear medium. In principle, any medium (like air) is nonlinear, but the nonlinearities are often so small that it is impractical to produce a nonlinear effect. If we want to demonstrate nonlinearity we use some special material like lithium niobate.
Generally even in a "strongly" nonlinear material producing interactions of this kind is fairly difficult and requires high power sources, careful choice of the wavelengths of the two sources, and careful alignment of the beams, to observe a strong effect.
There is one case I know of where nonlinearity of "ordinary" material plays a role: In wavelength-division multiplexed fiber optic systems, the nonlinearity of the glass fiber limits the number of WDM channels that can be effectively carried on a single fiber.