I applied for UK Visa, first time and it got rejected, now when I applied for the second time, I made a mistake accidently by not mentioning the refusal and I cancelled the application by calling them, however they told me that I will get the emails within the next few weeks but they told me that application is withdrawn and also given me the reference number. Is my application canceled? Now If I have to apply for the third time, should I write yes in false information section or how do I explain this?
They already have the information about your past refusals in front of them, of course. Although you withdrew the application, you did still make the application in the first place - you signed it and everything - so this is still likely a “false information” case, and not something you shouldn’t mention. Bringing it up gives you a chance to explicitly say it was a mistake and to point out that you voluntarily withdrew the application.
But do not take that advice; it doesn’t matter. Applying again after a second refused visa is not a DIY situation. You should be contacting a REPUTABLE immigration attorney (not a “service”, not an “agent”, but someone who is officially licensed to practice UK law). They will advise you on the best approach, which (depending on the circumstances of the previous refusal) will likely include “wait a few years first”.