I like her but she is clingy?

I am getting to know a girl; I am 23, she is 19.

We are about to start seeing each other over weekends and after work when possible etc.
I like her, she's very cute.

But she has told me, explicitly, that she doesn't like being put second at any time, and when I don't read her messages when she expects me to, she send more messages until I see them.

What do I do in this situation?
I like her but she is clingy?
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