I am wondering - is such connection has any sense - may it have some effect on generated sound?
Single wire cable from amplifier goes to speaker, and then set speeaker to bi-wire mode and connect speaker terminals with separate cable?
Let me illustrate this
No, this would be useless, you may as well leave it at the default connection. For bi-ampped speakers, the best you can do is maybe use un-used channels from a surround sound system to drive the second power connection if the receiver has options to set it up that way. Without a normal dual amp setup that is.
The principle behind biwiring a speaker is that if the speaker has separate crossovers for the tweeter/mid and woofers then extending the wiring back to the amp with long speaker cables, instead of the short jumper, minimizes the interaction between the drivers reducing intermodulation ditstortion. Depending on the design of the speaker it can be a waste of time or can yield some improvement. Worse case you can just double up and have a lower guage of wire which can give you a bit more control of the woofer.
Wire one speaker for biwiring and leave the other single wired, play a mono recording and use the balance control to compare them. Never know it to sound worse and most speaker cable isn't expensive.
There are some speaker cables that actually have different cables for the upper and lower range rather than just 4 of the same conductors. Usually expensive.