Yes, ""The author had been writing mostly fiction" is a participle phrase" is the wrong phrase.
A participle is a "verbal", meaning a word based in a verb, that functions as an adjective to modify a noun. There are present participles, which end in -ing, and past participles, which end in -ed, -d, -en, -t, -n, or -ne.
I saw my brother walking to the store.
walking to the store is the present participle phrase that describes the direct object, "my brother." walking is the participle, and to the store is the prepositional phrase that acts as an adverb in the participle phrase.
The purpose of the participle phrase is to modify and describe the noun. In this instance, the direct object "brother" is modified and described by "walking to the store."
Having removed his coat, my brother sat down.
The past participle phrase is "having removed his coat," which describes the subject "my brother."
In the phrase "The author had been writing mostly fiction" does not contain a participle phrase. You can break the sentence down into its grammatical parts.
"The author" = article and subject, "had been writing" = verb in past perfect progressive tense, "mostly" = adverb, "fiction" = direct object.
There is no participle phrase in this sentence.
The other four phrases are correct.
"I went to the store" contains a prepositional phrase: "to the store."
"Knitting sweaters is relaxing" contains a gerund phrase which acts as a noun: "knitting sweaters."
"I love to run long distance" contains an infinitive: "to run."
"Long novels with unpredictable plot lines are my favorite to read" contains a noun phrase: "long novels with unpredictable plot lines."