Whether or not the comma is necessary depends entirely on what meaning you want to convey.
Is from the opening lines providing additional information, or is its presence essential to the sentence?
Syntactically, because the comma is there (and how the rest of the sentence is constructed), what precedes it is a dependent clause, which makes the information nonessential.
From the opening lines, it's clear The Girl at the centre of these poems is damaged.
It's clear (and was from the start) The Girl at the centre of these poems is damaged.
It's clear The Girl at the centre of these poems is damaged.
But if the author had intended from the opening lines to be essential information, then the comma should not have been used.
From the opening lines it's clear The Girl at the centre of these poems is damaged.
The opening lines make it clear The Girl at the centre of these poems is damaged.
It's clear because of the opening lines The Girl at the centre of these poems is damaged.
Note that this may be a distinction that is lost in some interpretations. For instance, the comma may be viewed stylistically rather than syntactically.
In other words, the version of the sentence that has a comma could still be seen as providing essential information, with the comma serving only to indicate a pause rather than to indicate a dependent clause.
As for the use of that, it's mostly a style choice. It may or may not sound better to you or a reader, and in some cases its presence can help to more easily parse a sentence. (While in others it could be considered unnecessary or even distracting.)
Neal Whitman discusses this in the the blog post "When to Delete 'That'".
He says that "newspapers are often guilty of ignoring the difference between bridge verbs and non-bridge verbs and deleting a 'that' after verbs where it would sound better to leave it in."
He further says:
Sometimes, omitting a "that" after a non-bridge verb goes beyond being slightly awkward and can actually be confusing. Here’s an example from Bryan Garner’s Modern American Usage:
- Son acknowledges being a member of a minority . . . may have helped him turn his eyes abroad early.
The trouble here is that "acknowledge" can be a transitive verb. So when a noun phrase comes after it, such as "being a member of a minority," the reader might just take it as a direct object: “Son acknowledges being a member of a minority.” But whoops! The sentence keeps going, and the reader has to go back and reparse it. Garner calls this a miscue; sentences that produce miscues like this are called garden-path sentences.