Your question indicates that you might have a little confusion about put options and/or leveraging. There's no sense I'm aware of in which purchasing a put levers a position. Purchasing a put will cost you money up front. Leveraging typically means entering a transaction that gives you extra money now that you can use to buy other things. If you meant to sell a put, that will make money up front but there is no possibility of making money later. Best case scenario the put is not exercised. The other use of the term "leverage" refers to purchasing an asset that, proportionally, goes up faster than the value of the underlying. For example, a call option.
If you purchase a put, you are buying downside protection, which is kind of the opposite of leverage. Notice that for an American put you will most likely be better off selling the put when the price of the underlying falls than exercising it. That way you make the money you would have made by exercising plus whatever optional value the put still contains. That is true unless the time value of money is greater than the optional (insurance) value. Since the time value of money is currently exceptionally low, this is unlikely. Anyway, if you sell the option instead of exercising, you don't need to own any shares at all. Even if you do exercise, you can just buy them on the market and sell right away so I wouldn't worry about what you happen to be holding.
The rules for what you can trade with a cash instead of a margin account vary by broker, I think. You can usually buy puts and calls in a cash account, but more advanced strategies, such as writing options, are prohibited. Ask your broker or check their help pages to see what you have available to you.