As you correctly figured out,
for his girlfriend start with a preposition followed by a noun phrase. This is called a prepositional phrase (= PP): The head (= the element which determines the grammatical properties and the main meaning of the phrase) is a preoposition, and the noun phrase is a complement that is selected by the prepositional head.
But PPs are not the only elements that "extend otherwise basic clauses": Consider, for example,
He wrote a love letter yesterday
He wrote a love letter last year
He carefully wrote a love letter
He wrote a love letter alone (in his room)
None of these are phrases headed by a preposition, but nevertheless they serve a similar job as the examples mentioned by you: They modify a structure that is itself already a full sentence.
In general, as Atamiri already mentioned, such phrases are called free modifiers or adjuncts1. This is a very broad term for anything which in some way modifies something else .
In the case of clausal modifiers, this comprises most commonly PPs and adverb phrases (= AdvPs) (i.e. phrases that are headed by an adverb, such as
quickly), sometimes external adjectives (such as
alone, which is an adjective that in fact predicates over the subject rather than the verb phrase) or noun phrases (such as
last year, which is phrase that is headed by a noun and in turn modified by an adjective).
Oftentimes, modifiers that act on sentence-level like the above mentioned ones are also called adverbials, but this is discouraged by some syntacticians as "adverbial" might suggest the underlying part of speech were an adverb (like
quickly), which is not necessarily the case (as with PPs, for example).
But there are modifiers for other types of elements than just sentences: For example, adjectives/adjective phrases can serve as modifiers to noun phrases (e.g.
big shoes), and PPs can also modify noun phrases (e.g.
on the table in
the books on the table).
Other than so-called complements or arguments2, adjuncts are not selected/subcategorized for by other elements: There is nothing in the sentence which requires them to be there, and the sentence would remain grammatical and pertain its meaning if they were left away. They merely add some more local/temporal/... information to the structure they attach to and are thus modifying the sentence.3
In contrast, arguments/complements are grammatically required and contribute to the core meaning of the sentence: For example,
a love letter is an argument selected by the verb
night is a noun phrase complement selected by the head
at. Without them, the structure would be ungrammatical or drastically change its meaning4.
Last but not least, I should mention that the binary distinction "complements vs. adjuncts"/"arguments vs. modifiers" is not as simple and undisputed as it may seem from my exemplifications: Many people argue - for good reasons - that such a two-way distinction is obsolete and that there need to be more more finegrained levels of how an element may contribute to a sentence. But that is a bit out of the scope of this post.
TL;DR: "adjunct" or "free modifier" is the term you are looking for.
1 The distinction between "free modifier" and "adjunct" is not as straightforward; "modifier" is a bit more theory-independent in that "adjunct" is usually used in the context of phrase structure grammars, but "modifer" is also a more vague term that is more semantically than syntactically motivated. You'll find both terms in the literature, but in computational parsing, which often relies on dependency structures, "modifier" is probably the more common term.
2 The distinction between the two terms "complement" and "argument" is not always made accurately either; sometimes the criterion is that an argument needs to be something nominal, i.e. a phrase that semantically refers to concrete individuals taking part in some event, while "complement" is a more general syntactic term that may apply to any category, for example PPs which may be selected by verbs (such as
on the table in
John put the flowers on the table) without being a noun phrase.
3 Another way to check whether something acts as a complement or as an adjunct is that adjuncts may be iterated rather freely, while complements need to be present exactly once (i.e. they may neither be left away nor may there be more than one element occupying the same complement position):
He wrote a love letter (0 adjuncts)
He wrote a love letter alone (1 adjunct)
He wrote a love letter alone at night in his room with a fountain pen (4 adjuncts)
* He wrote (no verbal complement)
* He wrote a love letter an SMS (2 verbal complements)
The adjuncts to a sentence may be iterated practically unlimitedly - including the possibility of there being none at all -, while the complement position needs to be occupied no less than and no more than once.
4 Not a particularly nice example, but cf.
John drowned vs.
John dronwed Mary: Leaving away
Mary still keeps the sentence grammatical, but the role John takes in the action (in one he is actively causing the drowning to someone else, in the other he is the one undergoing the drowning) changes drastically - so
Mary must be an argument rather than an adjunct. In contrast, adding
at night to the love letter sentence doesn't fundamentally change the sentence's meaning or grammaticallity: It merely acts as a modifier; thus it is a free modifier or adjunct rather than a complement.