Will what work? Games? And at what resolution? Certain less-demanding "esports" type games might potentially be able to push 240fps on that hardware at reduced graphics settings, but for the vast majority of modern games I would say no, probably not even close at resolutions of 1080p and above.
Are you gaming on the 3400G and 1660 already? If so, getting a 240Hz monitor is not going to affect the frame rates that hardware can put out, assuming the resolution doesn't change. A higher refresh-rate monitor increases the number of times each second that the monitor refreshes its image, but the system's core components like the graphics card and CPU will determine how many frames get rendered and in turn fed to the monitor. So if you are getting, for example, 100fps in a particular game at 1080p on your 3400G and 1660 now, a 240Hz monitor at the same resolution will not change that.
However, if you only have something like a 60Hz screen now, then only 60 of those frames would be getting displayed each second. So, in the example of getting 100fps, 40 of those frames would not be getting displayed by the screen. So, a high-refresh monitor could allow all of those frames to get displayed, resulting in smoother motion, even if you don't manage to match the screen's maximum refresh rate. Of course, unless you have games that can already consistently push more than the 144Hz or so that is more common for high-refresh screens, paying a significant amount extra for 240Hz would probably be a waste.
In order for a 240Hz screen to provide any real benefit, you would need to already be pushing higher frame rates than what a 144Hz screen can display. A graphics card upgrade would likely make the most difference to frame rates, but there is a major shortage of graphics cards this year, resulting in prices that can be around double MSRP, so it's not exactly a good time to be upgrading graphics hardware.