This is because in CMYK images where you have two distinct colours, a gap can appear when the image is printed. Basically, there is no black ink in your red text (which is hardly surprising!).
What you need to do is employ trapping when outputting your CMYK image, to help close up the gap. Trapping is basically an outline around the text to help mitigate this problem. When printing, this trapping will also help if there is a slight misalignment between the plates on the printing press.
It's really easy in Photoshop with a file like yours to trap text against a background.
Before outputting the finished CMYK image for print - click
Trap. 1 px should be enough. This will flatten all layers and rasterize everything, so make sure you don't overwrite your original image!
Examples: Untrapped (left) and trapped (right)