How does the Catholic Church reconcile the holy Catholic Church with (the) Pope(s) asking for forgiveness?

Nothing in Catholic doctrine suggests that the Church is absent of sin, nor that its members or clergy are free from the need for forgiveness. To the contrary, it explicitly teaches in various formats that all people are sinful.

From the Catechism of the Catholic Church:

All men are implicated in Adam’s sin, as St. Paul affirms: “By one man’s disobedience many [that is, all men] were made sinners”: “sin came into the world through one man and death through sin, and so death spread to all men because all men sinned....” The Apostle contrasts the universality of sin and death with the universality of salvation in Christ. “Then as one man’s trespass led to condemnation for all men, so one man’s act of righteousness leads to acquittal and life for all men.” (CCC 402)

No exception is given for clergy of any level -- nor the Pope. All men (and women) are stained with sin. All are sinful in nature.

Consider your verse from Ephesians again, in a fuller context.

²¹ Be subordinate to one another out of reverence for Christ. ²² Wives should be subordinate to their husbands as to the Lord. ²³ For the husband is head of his wife just as Christ is head of the church, he himself the savior of the body. ²⁴ As the church is subordinate to Christ, so wives should be subordinate to their husbands in everything. ²⁵ Husbands, love your wives, even as Christ loved the church and handed himself over for her ²⁶ to sanctify her, cleansing her by the bath of water with the word, ²⁷ that he might present to himself the church in splendor, without spot or wrinkle or any such thing, that she might be holy and without blemish. ²⁸ So [also] husbands should love their wives as their own bodies. He who loves his wife loves himself. ²⁹ For no one hates his own flesh but rather nourishes and cherishes it, even as Christ does the church, ³⁰ because we are members of his body. ³¹ “For this reason a man shall leave [his] father and [his] mother and be joined to his wife, and the two shall become one flesh.” ³² This is a great mystery, but I speak in reference to Christ and the church. ³³ In any case, each one of you should love his wife as himself, and the wife should respect her husband. (Ephesians 5:21-33)

The implication in Ephesians isn't that the people of the Church, though holy ("set firmly apart" for God's plans), is pure. Rather the opposite is stated, that the Church is blemished by sinful people and is being actively purified by the Word of God (Jesus Christ).

This understanding of the Church is supported by Fr. Barron, a prominent Catholic Priest and Catechist, with specific regards to the sexual abuse scandal. His interpretation of the scandal, and sin in general within the Church by extension, is available on Youtube.

The Church is holy, but it is an institution which is populated by human beings, who can and do sin. This can be seen in the first few chapters of the book of Revelation where the Angel catalogs the sins of the seven churches.

In his apology, the Bishop of Rome is merely acknowledging the sinfulness, individually and collectively, of members of the church, even though the Church itself, as the Body of Christ is preserved Holy by the Spirit of God.

Perhaps this is a novel analogy. In my "day job", I work with shipping pallets, those devices constructed of wood (and other materials) which are used to unitize freight to make for easier handling. Now, some constituent parts of some wood pallets are defective: incomplete, broken, or rotten. But a Master Carpenter can take broken, defective, incomplete, and rotten members, and construct a pallet which, notwithstanding the apparent inadequacy of the parts, still performs the function it is intended to perform. So it is with the Church. The Master Carpenter takes the materials at hand, and combines and attaches them in a way to make a creation that admirably performs the mission to which it is called because the strong parts of one member compensate for the weak parts of another. Or, to put it another way, a fully functional pallet is made from less than functional parts.

In just the same manner, the Master Carpenter takes human sinners, and uses them to make a Holy Church.