background() can take an image as parameter.
You need to make your size() a bit bigger to take in account the borders, and to offset your sketch, ie. not to draw starting at 0, 0, but at x, y which is the position of the upper-left internal corner of the frame.

Alternatively you can set a PNG at the end of the draw()

Fantastic. Thank you both.

One other question that's related to the drawing code.

When I run my programme, the first line is automatically drawn from (0,0) to the cursor point rather than just starting at the cursor's coordinates. Again, it's probably something really simple that I'm just missing.

Theoretically, we cannot answer your last question, as it is too generic, without code... Anything can happen.
Fortunately, I recently had a similar problem, which can (or not) be related to your.
I suppose you draw from pmouseX to mouseX or something like that. You have to add a conditional checking the initial value and rejecting it.

My remark still apply. If prevValX/Y are never negative, initialize them at -1, and if they have this value, don't call line().

 float potValX = 0;       
 float potValY = 0;       
 boolean button = false;      
 float easing = 0.5;
 float prevValX = -1;
 float prevValY = -1;

 void draw() {


float targetX = potValX;
x += (targetX - x) * easing;
float targetY = potValY;
y += (targetY - y) * easing;

if(prevValX!=-1 &&prevValY!=-1){
line(x, y, prevValX, prevValY);
}

prevValY = y;
prevValX = x;

}

I've tried the above and it still draws the line at the start but from about half way between (0,0) and (126,119)

I initialised prevValX and prevValY as 126 and 119 respectively and it solved the problem. Thank you very much for your help. :)