Can an HDMI cable carry a mouse a keyboard signal?

No, HDMI cables don't support this functionality.

HDMI does have a backchannel (CEC) which in principle could be used to carry mouse and keyboard signals, but I don't know of any hardware which actually uses it for that. You may be able to use it that way in hardware you built yourself, but it's extremely unlikely your monitor supports it.

Monitors sometimes have an integrated USB hub for mouse and keyboard, and require an USB cable back to the computers.

Unofficially it seems, from this CKLau KVM switch that I have purchased. No affiliation.

Update

Yes, this does work for this device: USB mouse and keyboard data is carried along with HDMI from the KVM's HDMI output to the device where it has a breakout of HDMI and USB plugs that plug into the device.

The keyboard and mouse plug separately into USB inputs at the KVM.

Presumably the implementation in the KVM to carry USB data could be making use of of the CEC as this other answer here mentions: https://superuser.com/a/1155529/21353 Either that or an unofficial proprietary out-of-band method.

In conclusion, I'm not keen on this device for what seems to be a 'hack' / shortcut.

Also because on a separate note it does not carry the monitor's EDID information to communicate the display capabilities to the end device. I see 800x600 default Windows display but when plugging same device into a HDMI cable directly between it and the monitor, the native 1920x1080 resolution of the monitor is made use of.

So I wouldn't recommend KVMs that unofficially support USB over HDMI.