I have one fallopian tube due to a cyst and very irregular periods. Anything I can do to help my chances of conceiving?

Get help. See a doctor who specializes in fertility medicine; either an rei or a obg with expertise in the area. The combination of irregular cycles with a missing tube may make it quite difficult, maybe impossible, to conceive without guidance, but chances can still be very good with help.
Q for REI. This is a complex question about ttc. I will refer it to some of our leading fertility doctors on healthtap. Thanks for asking.