Which Xibo api can be used to add webpage widget in a playlist for a layout?

You will need 1.8 CMS and use 1.8 API, that being said, adding modules to a playlist is not documented yet.

The call would be POST /playlist/widget/:type/:id

type - webpage
id - playlistId

as a parameter you would need to pass uri, modeId, duration*

uri - URL(link) to your webpage.
modeId - open natively/best fit/manual, you will want to pass 1 as this parameter’s value
duration* - there is a bug it does require code change to work without duration provided, without cms changes, you will need to provide duration parameter with the call.

something like this:

please look at my API guide for more information if needed - 1.8 API Introduction