You can use "Save as" to save the file as .PNG, and to decrease the size you can lower the resolution by going to Image > Image Size.
Photoshop and Adobe applications in general are pretty abysmal at optimizing PNG images without a lot of manual intervention. I use Color Quantizer (free tool, Color quantizer ) to achieve great quality versus very small file sizes with a few clicks. Get the right tool for the job instead of fiddling around with Photoshop.
Also, the old Save for Web function is absolutely terrible at PNG optimization: it can't even handle basic 256 indexed fully transparent PNG export. Avoid like the plague.
See below: 97kb, 512 colours. Color Quantizer.
Or this one, 51kb, 128 colours, dithered, masked edge to preserve edge anti-aliasing (CQ features a brilliant mask tool to control precisely which colours ought to be preserved as much as possible). Color Quantizer has many other setting to control quality, the type of dithering, control whether gradients or details ought to be preserved, and many more.
Thanks everyone for your replies, I will give them a try. I didn't mention in my original post but the dimensions of the image are also restricted to 1080x1920 px so I don't believe I can change the image size.