You can't redraw to the screen with out calling background(). Otherwise the stuff just builds up. Without seeing your code I'm guessing that you're running through an array of images and putting them on screen every loop of draw().

I think one solution might be to save the image of the applet in a PImage and then use that as your background. Then when you need to make your changes, only draw the couple of images you're moving about and save the screen again.

Again my psychic powers are a bit rusty but here's how to save the screen:
Code:

PImage saveMe;
void setup(){
size(200,200);
saveMe = new PImage(width,height);
rect(20,20,20,20);
loadPixels();
saveMe = get();
}
void draw(){
background(0);
image(saveMe,mouseX,mouseY);
}
 

I've knocked up a demonstration of Processing chucking up 8 * 6 PImages a frame. It doesn't seem to run that slow. Could you say how you've done it differently to the example I've demonstrated? If you could post the offending bit of code it would help a lot - you might be able to use my "buffer filler" to avoid referring to mystery images.
Code:

PImage [] strip;
int offset = 0;
int stripSize = 100;
void setup(){
  size(800,600);
  smooth();
  stroke(255);
  //create a buffer of images of a spinning stick
  //so principle can be demonstrated from pure code
  strip = new PImage[width / stripSize];
  for(int i = 0; i < strip.length; i++){
    strip[i] = new PImage(stripSize, stripSize);
    background(0);
    float x = 50 + cos((TWO_PI / strip.length) * i) * 50;
    float y = 50 + sin((TWO_PI / strip.length) * i) * 50;
    line(50, 50, x, y);
    loadPixels();
    //"g" is a PImage that holds the image of the applet
    //copy() should be able to work with out it there but on 98 it doesn't
    //I'll try to remember to post a bug
    strip[i].copy(g, 0, 0, strip[i].width, strip[i].height, 0, 0, strip[i].width, strip[i].height);
  }
}
void draw(){
  for(int i = 0; i < height / stripSize; i++){ 
    for(int j = 0; j < strip.length; j++){
	image(strip[j], ((j + offset) % strip.length) * strip[j].width, i * strip[j].height);
    }
  }
  offset = (offset + 1) % strip.length;
}