Based on many SO questions from the past ...
... The answer is probably "No, this is not possible."
I would suggest, however, that you missed an important step in your development process. The problem you are facing is that you want to allow mobile users to enter text (in particular URLs that are a type of pain) from other sources into the text box on your site painlessly. Take a step back and look at other possible solutions to this problem that are not related to Javascript access to the user buffer.
In fact, the “solution” that you are currently trying to implement does not even help the user, because the “click to paste” button is only useful for the mobile user if they have already copied something to their clipboard, and if it succeeded, then, apparently, they already know how their own copy / paste functions of their browser phone work, so they don’t need your button. There are two necessary steps in the user stream you describe: copying the URL from another source and pasting it into your site. Even if what you are requesting now was possible, it could only help you make the second step easier for the user.
Instead, I suppose you should check how to copy and paste work with various available brands of smartphones and their browsers. If any of them use obscure or opaque interfaces that you are worried about that your users may be unfamiliar with, you have a button next to your text box that is only visible to users of these platforms that give them platform-oriented instructions on how to how to use their native copy and paste functionality.