What article should I use here?
"Scored 2 goals in the second half of his last game";
or "Scored 2 goals in a second half of his last game";
or without any? "Scored 2 goals in second half of his last game".
The first is correct:
Scored 2 goals in the second half of his last game
The reason is that we use the before superlatives (like most) and ordinal numbers (first, second, third, etc.).
a. The most-controlled therapy yielded the best results.
b. The first studies were conducted in early 1993.
c. The last security conference was termed a success,—John M. Swales, Christine B. Feak: Academic Writing for Graduate Students
It is useful to learn the rules about superlatives and ordinals as a single rule because the -st ending applies to most superlatives and the ordinal number first.