There are no permanently "missable" puzzles in Professor Layton and the Spectre's Call (EU) / Last Specter (US). The game allows you to find everything in a single playthrough, but some are easy to overlook.
After credits, you can freely roam to clean up. The "Puzzle Master's House" (bonus puzzles) unlocks after solving all main puzzles.
Tip: If truly stuck, consult a puzzle location guide (e.g., Professor Layton Wiki) for the specific missing number.