In that you're saying the ends are frayed, I am assuming you are using either a silica or ecowool wick. Neither one will burn, and you can dry burn them in place. Taking them out you can rinse them in water and heat them with a torch or a gas stove flame (my favorite), until it is red hot and it will be as clean and white as new. Or you can source replacement ecowool at most vendors, a quicker and easier solution.
To re-thread, one way you can try is to put a drop of your eliquid on the end to wet it and twist it to a point.
What I feel is easier is using a helper. I use a piece of white teflon plumbers tape, it's available at the hardware store, about a dollar for a roll that's a lifetime supply. I take a small piece of the teflon, enough to cover the frayed end and extend a 1/2 inch over, twist it tight around the wick and the end becomes a tiny point that is easy to thread through the coil. It allows you to grab it and pull the wick through the coil on the first try.