I am thinking about using the Mann Whitney U test over Student's classic t-test. But I was warned that I'd lose power and would require a higher sample size to compensate.
I don't understand: Why does using a non-parametric test decrease power?
The reason that parametric tests are sometimes more powerful than randomisation and tests based on ranks is that the parametric tests make use of some extra information about the data: the nature of the distribution from which the data are assumed to have come. However, their power advantage is not invariant, as it is often minimal but sometimes they have less power.
See pages 96 and onwards of David Colquhoun's old but still golden textbook Lectures on Biostatistics. It is available as a free pdf here: http://www.dcscience.net/Lectures_on_biostatistics-ocr4.pdf
Non-parametric tests are usually almost as powerful as parametric tests in the circumstances where the parametric tests are appropriate. However, in circumstances where the parametric test may not be appropriate because its assumptions are too badly violated, the non-parametric test may be more powerful.
Generally speaking there's a lot to be said for the Mann-Whitney
It doesn't, generally. In many cases, quite the opposite.
If the assumptions of the t-test hold perfectly, and the nonparametric test you use is the Mann-Whitney, then you lose a tiny amount of power$^\dagger$, because the t-test is the most powerful test at the normal under a location-shift alternative. (The t-test uses all the available information in the sample, if the assumptions hold - equal variance, normal distribution, independence, etc. But if you don't have normal distributions, it doesn't; and in many such cases the Mann-Whitney actually makes a more efficient use of the available information)
And even if you were exactly at the normal, the power loss is quite small (in large samples, it corresponds to needing 4.7% more observations to get the same power ... less than one in 20).
$^\dagger$ (There are other nonparametric tests that don't lose power against the t-test at the normal, but that doesn't mean the Mann-Whitney is a bad choice for a test of location shift, even if you're confident you have a population distribution close to the normal distribution.)
[This argument would be like arguing against buying very cheap insurance on the basis that if nobody was ever involved in an accident it would be cheaper.]
Do you know that the data were drawn from a normal distribution? Otfen it's possible to tell -- even without looking at the data -- that they can't be (often one can tell simply by knowing that the variable is bounded; if it can't be negative, for example, it can't actually be normal). And if the distribution that the data were drawn from is even a little heavier tailed than the normal, the t-test is likely to be less powerful, not more; and it can be much less powerful
(In very small samples other considerations than power come into play and I sometimes argue for a parameteric test then, even though they can be sensitive to assumptions)