New Zealand Law
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This is a serious breach of your employment contract, and could lead to you making a personal grievance claim. It does not entitle you to take leave however, although you could tell your employer you need sick leave for the stress this has caused. So you can ask for leave as a remedy for the breach of contract, but there is no requirement they must give this.
No doubt the distress of not getting paid on time when you have your own bills to meet is serious. This would mean that they could be ordered to pay damages and ensure you are paid properly in future. If there are any direct losses such as interest charges by your bank or late fees you can claim these too. I suggest you start by telling them this is serious, and that you will bring a claim unless they sort this, and that you want some compensation for the stress. Then see what the management say. They may realise then that there is a problem