I had given up on this and last night it came to me in a dream. No kidding!
Simple solution really. I just put the full search url in the form plus the added term in the input and it works.
And a search input for flowers shows results for red flowers.
<input type="text" name="site_search=red+" />
<input type="submit" value="Search" />
Only thing is the url is a bit messy looking like this...
but it works.