That story about Jesus making a paste with his spit will likely sound odd to those who have been exposed to little more than the theology of Jehovah's Witnesses, but that is because Jehovah's Witnesses are so religiously isolated and academically ignorant in critical theology.
In Christianity there is a basic fundamental belief to the majority of almost every Christian. It is called "sacramentality." If all you have ever known is Watchtower religion then this passage likely weirds you out.
It will be too long and drawn out to fully explain what a sacrament is to you because it depends if you understand Christianity's theology on grace. As a Jew myself, I am not here to give you a lesson on becoming a Christian anyway.
In a nutshell, Christians believe that God mercifully considers the needs of humans to see, feel, and touch something while invisible spiritual actions are taking place. So God uses sacraments, everyday physical objects or persons, and endows them with the miraculous to such an extent that these objects or persons become the physical representatives of the invisible reality.
To illustrate: The hem of Jesus' cloak acted as a sacrament for the woman who was suffering from a hemorrhage. (Luke 8:44) Peter's shadow was a healing sacrament at one time. (Acts 5:15) Articles of clothing and handkerchiefs from Paul were also sacraments used to heal people. (Acts 19:11-12) Today most Christians practice two sacraments, that of baptism and Holy Communion or the Lord's Supper. The objects here are water and bread and wine or juice of grapes, respectively.
The paste Jesus made in your example was a sacrament. Even Jesus himself is viewed by Christians as a sacrament. Since God cannot be directly viewed and touched by humans, God came as the Incarnation as God the Son, someone humanity can see, touch, and understand in human terms.
Again this is Christianity's take. It doesn't represent my view as a Jew since Jews don't accept Jesus as Messiah. But it should help you understand what that mud/paste healing meant.