I feel that either of 4-bromo-3-chlorocyclohex-1-en-5-yne or 3-bromo-4-chlorocyclohex-5-en-1-yne is correct.
You have already correctly identified all the parts of the name and their correct order, only the numbering of locants is wrong. The most important simplified criteria for the numbering in such cases are:
- lower locants for the principal characteristic group that is expressed as suffix
- lower locants for multiple bonds (first to double bonds if there is a choice)
- lower locants for prefixes
- lower locants for substituents cited first as a prefix in the name
The corresponding actual wording in the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book) reads as follows:
When several structural features appear in cyclic and acyclic compounds, low locants are assigned to them in the following decreasing order of seniority:
(c) principal characteristic groups and free valences (suffixes);
(i) low locants are given to hydro/dehydro prefixes (…) and ‘ene’ and ‘yne’ endings;
(ii) low locants are given first to multiple bonds as a set and then to double bonds (…);
(f) detachable alphabetized prefixes, all considered together in a series of increasing numerical order;
(g) lowest locants for the substituent cited first as a prefix in the name;
The compound that is given in the question doesn’t have a principal characteristic group that is expressed as a suffix. Therefore, Rule (c) is not relevant in this particular case.
Low locants are assigned to the multiple bonds according to Rule (e). Since there is a choice between the ‘ene’ ending and the ‘yne’ ending, the low locant is assigned first to the double bond, which results in the name cyclohex-1-en-3-yne.
Next, low locants are assigned to the other substituents as a set according to Rule (f). However, the numbering of the ring is already given by the previous step; therefore, the only possible numbering is 5-bromo-6-chlorocyclohex-1-en-3-yne.