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In relation to obtaining POA approval to manufacture an article classified as ETSO, is it necessary to have applied for a DOA approval in addition to the ETSO authorisation for the article?

As can be seen on the relevant EASA website (ETSO FAQ), you must be, or have previously applied for, the ADOA (or DOA, if it is an ETSO article which is an APU) and the POA.
 
The DOA/ADOA and ETSO are held by EASA. The POA is handled by AESA as the Spanish Aviation Authority. Therefore, all 3 approvals have to be requested at the same time (approximately), from the respective Authority.
 
A.133 (Eligibility) is essential to remember in relation to the POA:

What is an EASA aircraft and a non-EASA aircraft

Non-EASA aircraft, are those types of aircraft by which they are excluded from EASA Regulation 2018/1139. Among these types are ultralight motorised aircraft called ULM, Historical, Experimental, Amateur Construction Aircraft. Also excluded from the EASA regulations are those aircraft that are used in the use of military operations, patrol boats, police, etc. For more details of which aircraft are excluded, you can consult Regulation 2018/1139, in its Article 2.3(a) and Annex I.