How to erase a shape without redrawing the entire surface?

On the screen below I have the image with the painted translucent rectangle and with the painted opaque rectangle - which cuts the area from translucent rectangle. Now it works because I redraw the surface and the translucent rectangle each time in the cycle, but working with slow computers this redrawing causes glaring. If I stop redrawing all this stuff, I get this result, but the glaring disappears:

The commented lines in code below cause the result at the picture

I want just to erase the the translucent rectangle without redrawing the entire surface. I want to dynamically select the area (without redrawing the entire surface each time) for making a screenshot like on the Windows 11

  void OcctGtkViewer::buttonclick(int *x, int *y, int *width, int *height)
  {
    cairo_surface_t *surface = cairo_xlib_surface_create(xdisplay, xroot,    DefaultVisual(xdisplay, scr), DisplayWidth(xdisplay, scr), DisplayHeight(xdisplay, scr));
    cairo_t *cr = cairo_create(surface);

    const char *backgroundImagePath = "/tmp/test.png";
    cairo_surface_write_to_png(
     surface,
     backgroundImagePath);

    cairo_surface_t *surfaceTmp =    cairo_image_surface_create_from_png(backgroundImagePath);

 cairo_set_source_rgba(cr, 0, 0, 0, 0.2);
 cairo_rectangle(cr, 0, 0, DisplayWidth(xdisplay, scr), DisplayHeight(xdisplay, scr));
 cairo_fill(cr);

 cairo_surface_t *surfaceTmpp =  cairo_image_surface_create_from_png(backgroundImagePath);

 printf("select an area\n");
 while (1)
 {
   event.type == MotionNotify && isPressed == true)
   {
     int tmp_x = xevent.xmotion.x_root;
     int tmp_y = xevent.xmotion.y_root;

     cairo_set_source_surface(cr, surfaceTmp, 1, 1);
     // cairo_paint(cr);

     cairo_set_fill_rule(cr, CAIRO_FILL_RULE_EVEN_ODD); // The default is  CAIRO_FILL_RULE_WINDING.

     // cairo_set_source_rgba(cr, 0, 0, 0, 0.2);
     // cairo_rectangle(cr, 0, 0, DisplayWidth(xdisplay, scr),  DisplayHeight(xdisplay, scr));

     cairo_rectangle(cr, init_x, init_y, tmp_x - init_x, tmp_y - init_y); // set rectangle
     cairo_fill(cr);
   }
 }
}