fixing magsafe - possibly bad solder contact

I have Magsafe charger which apparently has problems with its connector or near connector because it sometimes refuses to charge and when I twist the cord little near the connector it works fine. Are therr any good guides for possible fix, is it difficult? I am not sure if it is the connector or the cord which is faulty.

So I've read. On YouTube there is one video where someone open the connector easily by holding the connector with pliers and then using flat head screwdriver to wiggle that white plastic thing off. I haven't managed to do so, partly because at the moment I can only find pliers with thin, "sharp" tip, can't get good enouh grip. If the fault is in the cord, I could maybe just cut off the bad part and solder pieces together? But I'm still unsure whether or not the problem is the connection between cord and the connector or a faulty in the cord itself.