How difficult it is to prove your intent depends on the facts unique to any case. For example, did you have burglars tools, did you have property that did not belong to you, did you make admissions to the police, did you have a valid reason for your presence, do you have prior convictions that can be used against you (usually only used if you testify but can be used if there is a particular M.O. involved. These are just a few. Your attorney, depending on their level of experience, should be able to tell you what arguments the prosecutor will use. I wouldn't say "most" of the time but I think they some do lie in an effort to help the case. Sometimes its those lies that result in an acquittal.
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Most Police don't lie most of the time. I suspect some lie often, but fortunately it isn't a large number. On the other hands thieves lie most of the time. How easy is it to prove "with intent to commit a crime therein" depends on a lot of factors which were not in your post, and should not be on a public message board. If for example, you were caught in a private home in the middle of the night when the owners were not there and you broke a window to get in, its easy. If you were caught in a public park during day light hours, its not.
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