Is there a way, for a early withdraw on my 401K to be considered a "hardship" on my taxes, if my employer doesn't specifically have a "hardship" choice?

[MindyB]

It depends on what the hardship was.  There are some circumstances, like certain medical expenses or a qualified disaster that'll help lower or eliminate the penalty. 

If not, you could consider the “emergency personal expense". The IRS defines “emergency personal expense" as an unforeseeable or immediate financial need relating to necessary personal or family emergency expenses. (Section 115 of the Secure 2.0 Act)

An emergency will allow you to make a withdrawal, still subject to tax, but without the 10% early withdrawal penalty.  You can take one financial withdrawal emergency per year, not to exceed $1,000. After entering your Form 1099-R, proceed through the interview screens until you reach "These Situations May Lower Your Tax Bill." In the box for Distributions for eligible emergency expenses, enter the lower of $1,000 or your withdrawal.

[NCperson]

@r-mill65 that "emergency" withdrawal only saves $100 in tax on the maximum $1000 withdrawal.  you still owe the tax; however, the 10% penalty on the $1000 (assuming you are under 59.5 years old) is waived.