According to Popper, a science without limits or imperfections or fixable flaws isn't science: it's metaphysics. In this sense, the end goal of science - the construction of a perfect mirror to reality - isn't just unrealistic: it's also unscientific.

Totally agreed. The power of science is first and foremost in the openness of its fundamental paradigm.

I wonder if Popper has any scientific philosophies about honesty and truthfulness.

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If you eliminate all the wrongs, you are most likely left with something resembling a right. I wish people would apply this simple fact to religions, when they're already saying each other is wrong!