Is {xy | x, y ∈ Σ∗ and x contains more a’s than y} regular?
Since the language does not specify what $x$ and $y$ are, you can always let $y=\varepsilon$, i.e., the empty string, then the language only requires $x$ to contain at least 1 $a$. So equivalently, the language is

$\{x\in\Sigma^*\mid x\text{ contains at least 1 }a\}$.

Now it is easy to build a DFA to accept this language.