Should I add insulation?

Last winter, I have noticed that one wall in my house is cold. I took part of the siding off and that's how it looks like. The black tarp is cracked in few places. The vinyl siding is attached to those vertical 1x2 boards. Would it be beneficial to add insulation (foam or styrofoam) between the vertical boards?

 

Where are you located ?

I've always seen vinyl siding attached over a foil faced foam panel instead of furring strips.

 

I am in New Jersey. Not sure if that matters, but this part of the house is an addition built in late 80s. Walls are pretty thin.

 

So just doing the one wall right?
They make 3/4 xps (foamular) board that you could cut and place between the furring strips. Cut them at a tight fit and you won't have to mechanically fasten them but it wouldn't hurt to run a bead of silicone where they meet the furring strip to help with air sealing. I think 3/4 is stated at R4. If you're in zone 4 (or even 5), from the table I found 3/4 is fine.
You could get rid of the furring strips and foam the whole wall if you wanted as 3/4 will keep all of your trim pieces in line but you'll just have to make note of the studs since you'll have to fasten both the foam and the siding.