In brief, syntactic phenomena are a set of syntactical arrangements or functions that may occur across a wide spectrum of languages. An example is Wh-movement, also called A-movement. Let's break the discussion down to definitions. Syntax, in brief, is word order and relationships in sentence structure. A phenomenon is defined as any remarkable observable occurrence, circumstance or fact (Collins English Dictionary). In simplest terms then, a syntactic phenomenon is a remarkable observed occurrence etc in sentence structure. In linguistics, these phenomena are set; there are standard accepted categories of syntactic phenomena.
One category is Wh-movement, or A-movement when speaking of this syntactic phenomenon in other languages. The Wh-movement phenomenon is that interrogatives begin with wh-words (which include how as an honorary wh-word: who, what, where, when, why, how, whose, whom, which, whither, whether etc). Further, the interrogative syntax follows a set word order of Object Verb Subject with an auxiliary verb or a form of do: What is (aux be) the name? Where does (do) he work? This syntactic phenomenon of word order and verb requirements in interrogatives also occurs in languages other than English.
Another one of the categories of syntactic phenomenon (i.e., a remarkable observed occurrence of sentence structure) is tag questions. Tag questions are added to (i.e., tagged onto) declarative statements and confirm the point being made through a negated or negation inverted interrogative: Statements may end in questions, mayn't they? He won't come to the party, will he? It is cold today, is it not? They couldn't very well go, could they? The syntactical phenomenon of tag questions uses modal verbs and auxiliaries.
As far as I understand, a “Syntactic Phenomenon” is the syntactic analysis and progress of the structure of a language, in this case perhaps, English.