Hi,
You didn't answer all my questions

, so I will try to have a general answer.
1. Statistical tests with a greater sample size (60) will have greater power to identify smaller effect sizes than tests with a smaller sample size (30).
If you want to get better intuition you may do a simple exercise. Run a two-sample t-test on two sets of numbers with some difference in the average.
(for example
group1: 2,3,2,3,2,3,2,3,2,3,2,3 and
group2: 3,4,3,4,3,4,3,4,3,4,3,4 see what p-value you got.
Now run the same test on half of the groups (
group1: 2,3,2,3,2,3
group2: 3,4,3,4,3,4)
2. Maybe If you have only boys or only girls you actually fix one IV, if this IV is relevant to the model the fixed model may not be as good as the wider model.
So the main question is why do you expect the poverty variable to be significant in any of the two models?