Is Holy Mother Church in any way connected to the Virgin Mary? If not, how about in mysticism?
This may help.
Lumen gentium
Vatican Council II, Dogmatic Constitution on the Church, 21 November 1964.
Mary, type or figure of the Church
By reason of the gift and role of her divine motherhood, by which she is united with her Son, the Redeemer, and with her unique graces and functions, the Blessed Virgin is also intimately united to the Church. As St. Ambrose taught, the Mother of God is a type of the Church in the order of faith, charity, and perfect union with Christ.[18] For in the mystery of the Church, which is itself rightly called mother and virgin, the Blessed Virgin stands out in eminent and singular fashion as exemplar both of virgin and mother.[19] Through her faith and obedience she gave birth on earth to the very Son of the Father, not through the knowledge of man but by the overshadowing of the Holy Spirit, in the manner of a new Eve who placed her faith, not in the serpent of old but in God’s messenger without waivering in doubt. The Son whom she brought forth is he whom God placed as the first born among many brethren (Rom. 8:29), that is, the faithful, in whose generation and formation she cooperates with a mother’s love.
The Church indeed contemplating her hidden sanctity, imitating her charity and faithfully fulfilling the Father’s will, by receiving the word of God in faith becomes herself a mother. By preaching and baptism she brings forth sons, who are conceived of the Holy Spirit and born of God, to a new and immortal life. She herself is a virgin, who keeps in its entirety and purity the faith she pledged to her spouse. Imitating the mother of her Lord, and by the power of the Holy Spirit, she keeps intact faith, firm hope and sincere charity.[20]
But while in the most Blessed Virgin the Church has already reached that perfection whereby she exists without spot or wrinkle (cf. Eph. 5:27), the faithful still strive to conquer sin and increase in holiness. And so they turn their eyes to Mary who shines forth to the whole community of the elect as the model of virtues. Devoutly meditating on her and contemplating her in the light of the Word made man, the Church reverently penetrates more deeply into the great mystery of the Incarnation and becomes more and more like her spouse. Having entered deeply into the history of salvation, Mary, in a way, unites in her person and re-echoes the most important doctrines of the faith: and when she is the subject of preaching and veneration she prompts the faithful to come to her Son, to his sacrifice and to the love of the Father. Seeking after the glory of Christ, the Church becomes more like her lofty type, and continually progresses in faith, hope and charity, seeking and doing the will of God in all things. The Church, therefore, in her apostolic work too, rightly looks to her who gave birth to Christ, who was thus conceived of the Holy Spirit and born of a virgin, in order that through the Church he could be born and increase in the hearts of the faithful. In her life the Virgin has been a model of that motherly love with which all who join in the Church’s apostolic mission for the regeneration of mankind should be animated.