Can I get rid of small white spaces after tracing?

I don't think I can attach a file here, so I will try to describe the issue.

I traced a pretty photo-realistic portrait. The result is good. But there are lots of small white dots or areas throughout the trace output. I figures that I should try and join surrounding areas but I get an error messages. I also tried gap options in live paint, but that doesn't seem to do the trick (perhaps I am not using it well, newbie here).

It occurs to me that unless I am not using the join tool or the gap options properly, which is perfectly possible, the reason these are not having the desired effect is that all those white areas are not defined areas at all but very small gaps. Could that be the case? So there's no point trying to "close" them, or join them, or pick them with the selection tool to fill them with colour, or to do that in Live Paint, because there is just nothing there. It's just gaps showing (in white because the background artboard is white). Could that be the case?

I am not sure why those gaps occur in tracing. And since they are distributed over many areas in which the colours differ, I can't place a colour background between the artboard and the traced output, as this will make all the gaps the same colour.

Have you come across a similar problem before and if so, would you be able to tell me if I can deal with it (during or after tracing) or if I have to live with it?

Thanks a lot

Best regards

Chris