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In Chiswell and Hodges Mathematical Logic the authors define a sequent as such

"A sequent is an expression
(Γ ⊢ ψ) (or Γ ⊢ ψ when there is no ambiguity)
where ψ is a statement (the conclusion of the sequent) and Γ is a set of statements (the assumptions of the sequent) ...There is a proof whose conclusion is ψ and whose undischarged assumptions are all in the set Γ.".

They then go on to provide the axiom as follows:

Sequent Rule (Axiom Rule) If ψ ∈ Γ then the sequent (Γ ⊢ ψ) is correct.

Upon further reading about what exactly it means to be a sequent wikipedia I understand that for $psi$ to be satisfied or "correct", every element of $Gamma$ must be true as their all linked by AND conjunctions.

What confuses me is that suppose that there is an element in $Gamma$ which is false then the whole antecedent is false according to the wikipedia definition by conjunction and then even though $psi$ exists and is true in $Gamma$ ultimately $psi$ would be false. Could someone please explain to me and help clarify how the axiom and the definition can both be true. Thank you for all of your help.

The authors are introducing the basic elements of the proof system.

As you said, the definition of correct sequent $(Gamma vdash psi)$ is :

There is a proof [according to the rules of the system to be specified] whose conclusion is $psi$ and whose undischarged assumptions [premises] are all in the set $Γ$.

When the semantics of the language will be defined [see para 3.5] the authors will intorduce the concept of semantic sequent : $Gamma vDash psi$, defined as :

for every $σ$-structure $A$, if $A$ is a model of $Γ$ then $A$ is a model of $ψ$.

The definition formalizes the informal concept of valid argument.

Then, they will prove the basic result [see page 87 : the Soundness Theorem of Natural Deduction for Propositional Logic] :

$Gamma vdash psi text { iff } Gamma vDash psi$.

Having said that, the rules of the proof system are the "rules of the game" that allows us to derive conclusion from premises.

It is obvious that if $psi in Gamma$, we can derive it from $Gamma$ and this is formalized with the (Axiom Rule) above.

What if $psi$ is false ? No problem: the move is "formally" correct but the argument is still valid because the case $psi$ false does not contradict the definition of valid argument :

the conclusion must be true whenever all the premsies are true.

In geenral, the reasoning applies if some elements of $Gamma$ is false; the (Axioom Rule) applies (because a premise can always be derived as conclusion) without contradiction.