You can bring extra kit.
There are a few prohibited items you cannot bring with you to CFLRS. I don't know these myself but I recommend all candidates find out what they positively cannot bring. Yes, your kit will undergo inspection on arrival for contraband.
You will travel in civy clothing, so you will obviously bring with you a warm jacket to a winter serial. You'll want to bring with you a change of clothes or two, but I encourage you to pack as though you were going for a short weekend. Evenings and CB weekends, you will be wearing either PT gear, combats, or coveralls. Personal electronics, to include cell phones, lap tops, music players, tablets, ebook readers, etc. may not be permitted by your course staff. Other personal items such as books, decks of cards, Swedish penis pumps, etc. are generally permitted. Be prepared to have various personal liberties curtailed.
A few things to keep in mind are time and space. If your course has a storage space, it is for the entire course, and not just for you. If your course does not have a storage space, you will have to stuff your kit into your issued kit bags, under your bed and/or into a coursemate's car. In sum, you cannot pack for every contingency and you cannot pack 6 changes of clothes for every weekend.
That said, you will be on course for a very long time. On completion of BMOQ, you will be immediately sent off to either EWAT, BMOQ-L/NETPO and/or DP 1. Pack what you will require to maintain your morale during this time, bearing in mind the space every thing takes and the possibility of replenishment during weekends and leave periods.