K_U_H,
What you’ve described is a “mis-configured” DNS. You need to repeat your A code for either the www.domainname for the domainname.
I’m not very familiar with DNS coding (and have to look at my DNS records every time before changing one) but this should have been done by the hosting server’s new account utility so my recommendation is that you NOT edit your DNS but send a request to your host to change your A record for the non-www’s domain to match the www.domainname’s record.
As for WP (?) looking for the www’d domain name, if that is a problem, then you can alter your .htaccess using mod_rewrite to require (or prohibit) www.domainname (an example of which is provided at http://dk.co.nz/seo.
Regards,
DK