As Ben’s answer says, there is nothing wrong with frequently repeated pronouns. Pronouns are ‘invisible’ and we rarely notice them at all.
What is important, especially if you feel like you’re using a lot of pronouns, is making sure that your pronouns are used clearly. If there are multiple people of the same gender involved, having many pronouns can lead to ambiguity if you’re not careful, and while frequent pronouns will fly under the radar, ambiguity is something readers almost certainly will notice.
Take the first two sentences of your quote as an example:
The girl held an immense love for her, as she was her mother, the apple of her eye[,] the center of her being. She wanted nothing more than to be just like her.
There are clearly two people involved here, but because they’re both female and both referred to as she/her, it’s hard to tell who is who. Perhaps you’d already introduced their internal relationship earlier on, in which case the reader might not be confused; but reading this snippet on its own, it’s hard to avoid having to backtrack and read again to figure out what exactly is going on.
We learn in the following sentence that one of them (presumably the daughter) is called Alexandria; for the sake of argument, let’s say the other is called Catherine. Now let’s replace the pronouns with names in two different ways to illustrate the ambiguity:
Alexandria held an immense love for Catherine, as Catherine was Alexandria’s mother, the apple of Alexandria’s eye, the center of Alexandria’s being.
Alexandria held an immense love for Catherine, as Alexandria was Catherine’s mother, the apple of Catherine’s eye, the center of Catherine’s being.
Both these readings are possible, and readers are likely to stumble and be unsure which is intended.
Additionally, there is the fact that the phrase the apple of someone’s eye generally dislikes ‘moving upwards’ in age. It refers to someone you’re very fond and proud of, someone you dote on, and it is almost always used of children (or grandchildren, nieces and nephews, younger siblings, etc.) or sometimes spouses – rarely of parents or other elders.
This means readers will intuitively expect “the apple of her eye” to mean that the daughter was the apple of the mother’s eye, rather than the other way around – but both readings above have the mother as the apple of the daughter’s eye. It is of course possible to use the expression in this somewhat unusual way, but you would normally want to make it explicit that that’s what you’re doing, to avoid misunderstanding; here, it goes in the unexpected direction implicitly, with both people involved referred to only with pronouns.
This will only add to the ambiguity already introduced in the first part of the sentence, to the point that it goes from being ambiguous to being downright confusing. Truth be told, I’m still not entirely sure who was intended to be the apple of whose eyes.
In cases like this, where repeated pronouns introduce ambiguity or even confusion, substituting a character’s name and/or rephrasing entirely becomes a necessity. Rephrasing won’t necessarily reduce the number of pronouns, but if you can rearrange the pronouns and mix them with names in a way that makes your text unambiguous, that’s all you need.
In this case, if the daughter is meant to be the apple of the mother’s eye, I don’t see a way to get rid of the ambiguity without a fairly thorough rewrite of the sentence – perhaps something like this:
The girl was Catherine’s daughter and held an immense love for her. She was the apple of Catherine’s eye, the center of her being, and she wanted nothing more than to be just like her mother.
If it is in fact the mother who is the apple of the daughter’s eye, just naming the mother will suffice (and perhaps adding a hint of the daughter’s age, just to dispel any lingering ambiguity):
The young girl held an immense love for her: Catherine was her mother, the apple of her eye, the center of her being. She wanted nothing more than to be just like her.
