can anyone tell me how to repair a broken beeswax candle?

Answer 1

If the wick is also broken, you can connect the parts but the candle will not burn past the break.
If the wick is not broken you can repair it so it will burn as if not been broken at all, but you might still see it break.

When the wick is broken, you can glue the parts, using melted wax for glue, the wax of the broken faces itself being good, just heat them up till the top layer gets soft and stick them together.
Keep a close watch on the candle when burning it, as the broken part of the wick will fall at one time and may fall outside the candle while still burning.

If the wick is not broken, get wax, (the parts fallen out, or from the top/bottom of the candle or from an other candle in wax) and warm them till they get supple, mostly the heat of your hands is enough, otherwise maybe a little while near a stove or central heating radiator.
You can now shape the wax, push it into the break and reshape the candle.
To fix it into place if it does not stick you need to warm it a little more.

If the candle is broken, wick not, but the space between the parts is to small to shape with wax, you can poor melted wax in the break, and glue it that way, or you can break a bigger part away to make the repairs.

Never have candle wax on a direct flame, you do not need to have it hot, just warm enough to let it shape or just melted.