I'm trying to make a cigarette. I made the burn area and the ash, but I can't get them to combine. Can you guys help me?

A mix shader should work just fine. Just make sure to use the same alpha factor that you used for the "burn mask".

You can combine them with mix rgb node or a mix shader node, for this task I'll reccomend you to use mix shader node just plug both the colors/textures in a principal or shader you think is more preferable and combine them, now for the next part you can use a gradient texture with mapping node and object coordinates and plug it into the factor input( you may need to inverse it for that just invert the order of sockets you have plugged into mix shader, the lower one in upper and upper one in lower), now you can use a color ramp to tighten the gradient as per your liking and use mapping node to move or rotate it, you can even add a noise texture in middle of mapping node and object coordinate to make a random coming ash on paper.