I've done the Mann-Whitney test and I'm not sure how to interpret the two-tailed p-value which was equal to 0.0336 and 0.0673. The test was used to find out how opinions, between men and women, differ in regards to genetically modified crops.
It's pretty clear that the first p value you cite is for a one tailed test and the second one for a two tailed test. Which one you should use depends on what your original hypothesis was, but, in the absence of a strong reason to think women would be higher on this, a two tailed test is appropriate.
Your last sentence, after the graph, makes no sense. You have the number of observations that you have.
Whether you should be using Mann Whitney at all is unanswerable here. I'd usually prefer to do some sort of regression with some control variables to get a better sense of what is going on.