First of all, if an image already has a profile, blindly assigning it any profile (whether Adobe RGB or sRGB or ProPhoto or anything else) is probably going to screw up the colors unless you know exactly what the original profile was, and why it was removed. If the image does not have a profile, the file is RGB mystery meat, and the originator of the file probably doesn't know crap about color management. Otherwise, they wouldn't put you in the awkward position of having to guess the correct colors in the file.
The thing you must understand is that RGB color numbers are meaningless without a color profile to put them in context. So RGB 255,0,0 signifies some kind of red, but how red, and exactly what hue of red? Only the associated color profile can answer that question. 255,0,0 in ProPhoto is much more saturated than 255,0,0 in sRGB. The numbers are the same, but the colors specified by the numbers are quite different.
Given that other images print reds properly, it's most likely that they contain the correct color profile (probably not Adobe RGB) so that a correct print can be made. But since I know nothing about how well your monitor is profiled, your color management settings in Photoshop, and your printing methodology, that's just a guess. It could even be a clogged ink nozzle in your print head.