My real reason for applying for polygyny is unmet intimate needs. Should I state this to the court?

If that is the real reason, state it honestly and decently, but be prepared for your wife to give her side too. Such matters are usually heard in private, and judges are used to assessing personal situations with care. Use language that protects your wife's dignity; avoid embarrassing details. Before filing, ask yourself: has this problem ever been discussed openly, referred to a doctor, or taken to counselling? Often the cause is exhaustion, health or an emotional wound that can be treated. If you have not tried that, start there. Perhaps what is needed is repairing the relationship, not adding a marriage. See etiquette of intimacy.