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:

• The production activity to be undertaken must be "appropriate" for approval under Part 21 Subpart G, as is the case for an ETSO item. In addition, and also essential to meet this criterion, is direct delivery to the final customer.
• You must own or have applied for approval of the design you wish to produce (which would be fulfilled by making the DOA/ADOA application).