EX-4.1




 Exhibit 4.1 

December 2, 2022 
 Holder of Warrants to
Purchase Common Stock issued on (i) November 25, 2020, (ii) January 25, 2021, (iii) June 1, 2021, (iv) December 28, 2021, (v) March 1, 2022, (vi) May 19, 2022, (vii) September 1, 2022 and
(viii) September 1, 2022 
 Re: Amendment to Existing Warrants 

Dear Holder: 
 Reference is hereby made to the
offering on Form S-1 (File No. 333-268251) on or about the date hereof (the “Offering”) by Baudax Bio, Inc. (the “Company”) of
shares of common stock, par value $0.01 per share (the “Common Stock”), and/or other securities of the Company (collectively, the “Securities”). 

This letter confirms that, in consideration for the Holder’s participation in the Offering and purchase of the Securities in the Offering
(the “Purchase Commitment”), the Company hereby amends, effective as of the closing of the Offering, the Existing Warrants (as defined below), by (i) reducing the Exercise Price (as defined therein) of the Existing Warrants to
$4.50 per share and (ii) amending the expiration date of the Existing Warrants to five (5) years following the date of closing of the Offering (the “Warrant Amendment”). The Warrant Amendment shall be effective upon the
closing the Offering and the satisfaction of the other terms and conditions referenced below. 
 “Existing Warrants” means,
collectively, the following existing warrants held by the Holder: 
  

	 	1.	 the existing warrants to purchase up to 7,234 shares of Common Stock of the Company at an exercise price of
$23.92 per share issued on November 25, 2020, as amended; 


  

	 	2.	 the existing warrants to purchase up to 7,358 shares of common stock of the Company at an exercise price of
$23.92 per share issued on January 25, 2021, as amended; 


  

	 	3.	 the existing warrants to purchase up to 6,013 shares of Common Stock of the Company at an exercise price of
$23.92 per share issued on June 1, 2021, as amended; 


  

	 	4.	 the existing warrants to purchase up to 5,143 shares of Common Stock of the Company at an exercise price of
$23.92 per share issued on December 28, 2021, as amended; 


  

	 	5.	 the existing warrants to purchase up to 48,246 shares of Common Stock of the Company at an exercise price of
$23.92 per share issued on March 1, 2022, as amended; 


  

	 	6.	 the existing warrants to purchase up to 14,404 shares of Common Stock of the Company at an exercise price of
$43.60 per share issued on May 19, 2022; 














	 	7.	 the existing A-1 warrants to purchase up to 142,858 shares of Common
Stock of the Company at an exercise price of $21.00 per share issued on September 1, 2022; and 


  

	 	8.	 the existing A-2 warrants to purchase up to 142,858 shares of Common
Stock of the Company at an exercise price of $21.00 per share issued on September 1, 2022. 


 The Warrant Amendment
is subject to the consummation of the Offering and the Holder’s satisfaction of the Purchase Commitment. In the event that either (i) the Offering is not consummated, or (ii) the Holder does not satisfy the Purchase Commitment, the
Warrant Amendment shall be null and void and the provisions of the Existing Warrants in effect prior to the date hereof shall remain in effect. 

Except as expressly set forth herein, the terms and provisions of the Existing Warrants shall remain in full force and effect after the
execution of this letter and shall not be in any way changed, modified or superseded except by the terms set forth herein. 
 From and after
the effectiveness of the Warrant Amendment, the Company agrees to promptly deliver to the Holder, upon request, amended Existing Warrants that reflect the Warrant Amendments in exchange for the surrender for cancellation of the Holder’s
Existing Warrants to be amended as provided herein. In addition, the Company shall file a prospectus supplement to the applicable registration statement in connection with the amendments hereunder on the date hereof. 

[Signature Page Follows] 

  
 -2- 












 IN WITNESS WHEREOF, the parties hereto have caused this agreement to be duly executed by
their respective authorized signatories as of the date first indicated above. 
  

			
	BAUDAX BIO, INC.
		
	By:	 	          

	Name:	 	
	Title:	 	

 Name of Holder: ________________________________________________________ 

Signature of Authorized Signatory of Holder: __________________________________ 

Name of Authorized Signatory: ____________________________________________________ 

[SIGNATURE PAGE TO BXRX WARRANT AMENDMENT
AGREEMENT]