The invisible cursor / big black X cursor sounds like a problem with Xorg, the display server.
Sadly, while I do not know of a fix or reason why a kernel update would break the cursor, there are a few things you could try to pinpoint if it's a system-wide or user profile issue.
- Does it happen with Marco window manager?
- Does it happen on the login screen or guest session?
- Does switching to a console (CTRL+ALT+F1) then back to X (CTRL+ALT+F7) show anything different?
- Do you have any proprietary drivers installed? I am presuming it's an Intel, in which case their display drivers are already part of the kernel.
The temporary workaround is to boot the older, working kernel by holding SHIFT at start-up, then choosing "Advanced options for Ubuntu", followed by the kernel before 4.2.0-27. If that solves it, then it's definitely a regression introduced in the new kernel, and you can save this as a permanent boot option for now if desired.
Might be overkill, if you know which update caused it, you can temporarily revert back to the older version of the affected package. There are a few ways to look up which updates you had installed.