If $4,200 is your mother's only income and she is single, she does not have to file a tax return. 

Tip: Even if she's not required to file a return this year, file one anyway if she wants to recover 2020 federal tax withheld from her paychecks.

• The TurboTax Free Edition, can be used to file simple returns with no additional schedules.  Upgrades and add-ons, such as TurboTax PLUS and Audit Defense, are available.

To review all the requirements to have to file a tax return, see Do I need to file a federal return this year?

She might have to file a tax return.  What kind of income does she get?  Did she get a W2 or a 1099Misc or 1099NEC?  

If she only got a W2 for under 12,200  she doesn't have to file except to get back any withholding taken out. If it is more or she got a 1099Misc or 1-099NEC for any amount you do have to file a return.

Be sure on her return you check the box that says someone else cam claim you That is VERY important!

Filing requirements for a Dependent
https://ttlc.intuit.com/community/children-dependents/help/do-i-need-to-file-my-own-taxes-if-i-m-a-d...

She has a small NYS Pension of $2415.00 a year, that's her only income

She has a W2

If her Pension (1099R) and W2 income are less than 12,200 she doesn't have to file except to get back any withholding taken out.   Are both the pension and W2 total less than $4,300?  If the total combined is more than $4,300 you can not claim her.   

It is less. $2415 to be exact. She is married and my father is filing as married / filing separately

It depends.  If your mother's income had withholding taxes, she may file income tax return to claim the refund.  You must also meet the following requirements to qualify as head of household:

1. Must have earned income. 

2. Unmarried at the end of the tax year. 

3. Supported over one half of the household cost. 

4. Must have a dependent living in the home. 

For more information, please refer to this link. 

Its a small pension ($2415), no withholding. She is married and husband is claiming file married/file separately.

The questions are:

1. Does your mother have to file a tax return?  No, as explained in answers above
2. Do you qualify as HOH if your mother is filing separately from her husband?  If you can answer the questions below, the answer is yes.

In order to claim Head of Household, you need to be:

• -unmarried or considered unmarried, meaning separated, divorced, or living apart from your spouse for at least the last 6 months of the year, 
• -pay more than 50% of the cost of maintaining a household for the year, in which a qualifying person lived for more than half the year.

Household costs include rent or mortgage payments, utilities, and other things specifically related to the residence.

One or both of your parents could be the Qualifying person in your household if you are claiming one or both of them as dependents.

From the IRS: 

You may be able to file as head of household if you meet all the following requirements. 

1. You are unmarried or considered unmarried on the last day of the year. 

2. You paid more than half the cost of keeping up a home for the year. 

3. A qualifying person lived with you in the home for more than half the year (except for temporary absences, such as school). However, if the qualifying person is your dependent parent, he or she doesn't have to live with you. See Special rule for parent, later, under Qualifying Person.

You can use the following link to go to the same Publication from the IRS to read more if you like:

https://www.irs.gov/pub/irs-pdf/p501.pdf

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