Without additional assumptions, the answer is basically no, not in any great generality.  The derivation of Black-Scholes requires that you can perfectly hedge movements in the option using a stock and a bond.  If the underlying stock price process has jumps, then you have jumps in the value of the option, and you can't hedge those jumps using only two assets.  (There is one exception — if the process is Poisson, then you can hedge the jumps, but as soon as you have jumps of more than one size then you're stuck.)

The additional assumption is some rule to determine how the option value jumps when the stock price jumps.  One rule is that the jumps are "idiosyncratic risk", and therefore are not hedged.  This is called the Merton jump-diffusion model.  There's plenty of material online about this model.  From a quick Google search, [these slides](http://www.math.nyu.edu/~benartzi/Slides5.2.pdf "Merton's Jump Diffusion Model: Peter Carr (Based On Lecture Notes by Robert Kohn)") ([Wayback Machine](https://web.archive.org/web/20170829225927/http://www.math.nyu.edu/~benartzi/Slides5.2.pdf "Merton's Jump Diffusion Model: Peter Carr (Based On Lecture Notes by Robert Kohn)")) look pretty good.