Mobile & Tablet versions look messy

Hi,

I have customised my site a fair bit and have used a background picture rather than a background colour for the search box (#main-searchbox {
background-image:).

Looks great on a PC but when I go to the mobile version its not centred as the picture is far too big and it look rubbish. Is there a way of styling the mobile version of the theme. I fear as its responsive it will merely share the same file and look different on different devices.

I wondered if there was a way of giving the mobile/tablet version of the site its own style line of CSS?

Or is the only way to make it look good to have a costly mobile app created?

Grateful for your thoughts,

Mark