Does adding glue inside a screw hole strengthen the connection of a wood screw?

Okay, there are several things at play here.

Screws are a mechanical fastener that cuts into the wood and holds because you’d need to either pull the plug of wood out also or pull hard enough for things to sufficiently deform for the screw to slip past. Now, in each case the screw will likely break before either occurs.

Glue does have mechanical properties once hardened (for the same reasons as above), but it is primarily known for its chemical bonds. Due to the nature of the materials involved wood glue, which will penetrate into the wood itself to a small degree, owes most of its strength to the mechanical bond (at least in this conversation).

So, if you just coat a screw with glue and screw the fastener in… You’ll pretty much have exactly the same values as the screw by itself.

If there is an existing hole/void that your putting the screw into the glue will aid as a filler. Its worth noting that most wood glues shrink as they dry so the interface between screw-glue and glue-wood is likely to be partial at best. But, if we assume that the whole void has been filled by screw and glue… You should have nearly the same value as the screw, but probably a little less. And that’s your best case scenario.

Basically, the glue would just act as a wood filler.