The client is paying you to develop software for them that they expect to just work. They are probably not sophisticated enough to understand that their environment might be an issue, and they probably assumed you - the professional programmer - considered the circumstances before you gave a price. If you ask for more money to debug it they are going to feel like you deliberately tricked them with your pricing.
I would debug it on their system. If it turns out there is something on their system that shouldn't be that is causing the problem that will cost you a lot of hours then you might say "This is extra because ____" but only if it takes a lot of time.
Use this as a learning experience and next time you bid on a project be sure to include minimum system requirements that the software will work on, and maybe give a price for the main project plus an optional extra item for more $ to "debugging on your system". Then if they don't take that you are covered. (I'd keep the price separate in case you're bidding against people who don't include debugging and come in with a lower price.)
As you bid on more and more projects you will start to learn the things you need to protect yourself from. Some things I cover in my proposal are:
The environment, when I have clients with multiple computers I say it's guaranteed to run on ONE system setup (their firm standard). IF it doesn't work on another computer, I'll trouble shoot but charge.
System changes - once I have it running on their system, if they change anything, (new OS, computers, new software that interferes, etc. that will be extra to troubleshoot. I can't control how their system might change!
Limited time for them to test. If they find a bug 2 years from now, even if it's due to my mistake, it's not reasonable for them to expect me to drop everything and fix it for free. (I will if I have time, but there are some people will not use the software for the longest time then start finding little things. You can't become their programming slave!
limited liability - if they think my software damaged something they can't get more $ from me than they paid for the software.
I offer an annual maintenance agreement, and if they turn it down, then they need to pay T&M for any fixes/updates, and I have a minimum X-hours for that type of work. Ironically, The type of people who turn down the maintenance agreement are usually the needy ones and you have to make them pay or they will suck you dry with requests/changes. But that is after you delivered a working product for them.
If you have to eat some hours on this, just know you've learned for next time. I think most of us have probably eaten a LOT of hours in the early days, not realizing all that would be involved.