Why do Priests put a piece of Hostia in the Chalice?
The symbolism of this act reminds us of the unity of the Church and the spiritual connection between each celebration.
It comes from the primitive church and a custom intended to mean unity with the Pope and the local bishop.
The symbolism of this act also points to the unity of the Body and Blood of Jesus in the Hostia.
Because both the Host and the Consecrated Wine contain the whole presence of Jesus: Body, Blood, Soul and Divinity.
It's a reminder that every priest's action at Mass has its roots in nearly 2,000 years of history and tradition.

“What is a priest! A man who holds the place of God–a man who is invested with all the powers of God. ‘Go,’ said Our Lord to the priest; ‘as My Father has sent me, I send you. All power has been given me in Heaven and on earth. Go then, teach all nations. . .He who listens to you, listens to Me; he who despises you despises Me.’ When the priest remits sins, he does not say, ‘God pardons you’; he says, ‘I absolve you.’ At the Consecration, he does not say, ‘This is the Body of Our Lord’; he says, ‘This is My Body.'”
-St. John Vianney.

According to Croatian tradition, when a couple gets married, the priest doesn’t tell them that they have found the perfect person. On the contrary! Instead, he says to them: "You have found your cross. It is a cross to love, to carry it with you, a cross that is not to throw away but to treasure."
In Herzegovina, the Cross represents the greatest love and the crucifix is the treasure of the house.
When the bride and groom enter the church on their wedding day, they carry a crucifix with them. The priest blesses the crucifix. When the time comes to exchange their vows, the bride puts her right hand on the crucifix and the groom puts his hand on hers, so that both hands are joined together on the crucifix.
The priest covers their hands with his stole as they exchange their vows, according to the rite of the Church, to be faithful to each other, in joy and in sorrow, in sickness and in health, till death do them part.
Then, instead of kissing each other, the bride and groom kiss the crucifix …More

tomorrowsworld.org/sites/default/files/booklets/pdf/gip-1.1.7.pdf
Not Catholic. But interesting reading material in PDF Format.

tomorrowsworld.org/sites/default/files/booklets/pdf/usp-15b.pdf
Not Catholic, but interesting reading material in pdf format.