It's hard to advise you on this, because offices are very different. Much depends on how big or small your office is, and how short staffed it may be.
I would think in the short term, as long as you are available for the scheduled relief day for your primary route, you should be fine. If you are short of RCAs and you will be expected to learn additional routes right away, your availability may be more of an issue. If they try to schedule you for a day where you can't be available, you will have to say no. Saying no too often can lead to problems, but again it depends on the specifics of your office. If your phone rings on a day where you are home with your child, don't answer it.
In the meantime, learn the job to the best of your ability. Don't give them additional reasons to think of you as a bad investment. If you do well, they will have a greater desire to work around your issues. Best of luck.