Was the Trinity severed during the crucifixion? When speaking about the crucifixion, I have heard from many Protestant's (mostly evangelical non-denominational Christians) state that the Trinity was severed or broken when Jesus was on the cross.
I don't understand the biblical basis of this or how it is possible. Was it miraculously restored when after the resurrection? According to Catholicism, what is the view on this? If they do agree that it was severed, what is the biblical verses that explain this?
Was the Trinity severed during the the crucifixion?
The answer is NO.  While I cannot find a specifically Catholic source for this answer, I did find a well-written article by Paul Copan a credentialed philosophy professor with degrees in theology and philosophy: Was the Trinity Broken at the Cross? who in turn relies on a 2012 book Forsaken by a professor of theology Thomas McCall.  I'm pretty sure that the explanation of how the Trinity was intact is acceptable to Catholics as well.
Here are several points from the article that addressed your questions:

The idea that there was a break within the Trinity at crucifixion is a modern development, most notable through the liberal theologian Jürgen Moltmann who wrote an influential 1972 book The Crucified God.

From the point of church history, church fathers and theologians—from the likes of Athanasius, Cyril of Alexandria, and John of Damascus to Peter Abelard, Thomas Aquinas, and beyond) did not take this view; they would have rejected what is a fairly recent theological development.


The Bible verse most likely to support a broken Trinity is Ps 22:1 which Jesus uttered from the cross in Mt 27:46:

And about the ninth hour Jesus cried out with a loud voice, saying, “Eli, Eli, lema sabachthani?” that is, “My God, my God, why have you forsaken me?”

but that is taking the verse out of context since later in Psalm 22 the psalmist turned to a note of triumph (v24):

For he has not despised or abhorred the affliction of the afflicted,
and he has not hidden his face from him, but has heard, when he cried to him.


Other Bible verses Matt 27:50 and Mk 15:37 seem to emphasize separation (Jesus uttered a loud cry and breathed His last / yielded up his spirit).  But notice that we are not told what he said.  Furthermore, parallel crucifixion accounts from Luke and John showed union between Father and Son:


In John 19:30, Jesus cries triumphantly, “It is finished!”
Jesus tells a criminal that he will be with him when he comes into his kingdom. Jesus says, “Today you will be with Me in Paradise” (Lk. 23:43). This doesn’t like a God-forsaken Son.
Jesus also calls on the Father while on the cross to extend forgiveness: “Father, forgive them, for they do not know what they are doing” (Lk. 23:34).
Jesus finally says, “Father, into your hands I commit My spirit” (Lk. 23:46) — again, no indication of a broken union—only that of a deep fellowship with his Father. We have a string of affirmations that the Trinity was united rather than broken, nor did “the Father turn his face away.”



&quot;Dividing God into parts is not possible metaphysically&quot;

By his very nature, God is necessarily triune; so the deep, unbreakable interrelationships between the persons of the Trinity cannot be broken. Indeed, the persons of the Trinity have mutually-indwelt one another from eternity (perichoresis [Greek]; circumincessio [Latin]), and they are necessarily bound up with each other. The persons of the Trinity are not detachable “parts” of God.
Furthermore, there is no opposition or lack of harmony between these persons. The Father does not “take out his wrath” on the Son. The saving plan of the Trinity before the Incarnation was mutually agreed upon by all persons of the Trinity from eternity. ...
Just as the Son is not forced by the Father to die but voluntarily lays down his life and takes it up again (Jn. 10:18), so the Father is lovingly and reconcilingly involved in the death of Jesus: “God [the Father] was in Christ reconciling the world to Himself” (2 Cor. 5:19). The triune God—not just the Father—is wrathful against sin, and the triune God—not just the Son—is lovingly involved in redemption. Love and wrath are not opposed to each other in God. Rather, wrath flows from the love of God.


&quot;We can still speak of a kind of “abandonment” without speaking of a broken Trinity or of the Father’s pouring out his wrath on Jesus—as though the Son of God is not wrathful against sin.&quot;


(for more explanation why Trinity was not broken, please refer to the article linked above)
Conclusion:

In the end, we do not have any rift within the Trinity. This is not biblical; it is not in keeping with the stream of Christian theology over the centuries; and it is utterly impossible metaphysically. Rather, we have a Triune God who is involved not only in creation (Gen. 1:1-2; Jn. 1:1-3), but also in the redemption of human beings.

P.S.: nathan.j.mcdougall's comment on penal substitution and Thomas McCall's defense of it while keeping Trinity intact is worth preserving:

The fact that particular Protestant formulations of the atonement (namely, Penal Substitutionary ones) ostensibly lead to either a rupture of Christ's human and divine natures, or else of the Trinity itself, is the basis for a Catholic argument against such models of the atonement. But generally Protestants would deny that such ruptures can take place, as Catholics would.