It could be either one, depending on whether "trying" is a noun (then you use "alleged") or a gerund, which is a verb form (then you use "allegedly"). This is because adverbs can't modify nouns, and adjectives can't modify verbs. The ambiguity of structure arises because "-ing" has multiple uses in English, and here, we might be dealing with the "-ing" that derives nouns from verbs or the "-ing" that accompanies gerunds, which are verbs.
Some English speakers might not accept the adjective "alleged" here, because they could be missing a lexical noun "trying". This is a difference between inflectional endings, like the ending in the gerund (which does not change the part of speech), and the derivational ending in the noun "trying". Derivational forms are not basic to the structure of the language -- individuals may differ in whether they are acquainted with specific words in the lexicon.
What is going on is clarified if we change the example so that "trying" has a direct object, since nouns do not take direct objects:
Alice's trying ice cream had left Bob wary of any gourmet foods she might suggest.
Now, you can't use the adjective "alleged".
It depends on how you think about grammar. If you like the adverb as a traditional part of speech, then sure, it's an adverb. If you analyze grammar and syntax based on function, then you might agree with some linguists that home is an adverbial / prepositionless prepositional phrase, or you might agree with other linguists that it is a preposition. I'll lay out each of the cases in detail.
Adverb?
First, dictionaries will tend to classify this usage of home as an adverb because lexicographers are being asked to put each single word into a part of speech, and they aren't doing much sentence-level analysis. So they pick one of the parts of speech inherited from 18th and 19th century grammarians, "adverb," which is really a catch-all for words that seem to modify adjectives, verbs, clauses, prepositions, and other adverbs (ThoughtCo, "What Is an Adverb?"). So the Oxford English Dictionary has an entire entry, "home, adv.," including this entry describing the usage in "He is home":
1e. Without verb of motion. Arrived at one's house, neighbourhood, or country after a period of absence. Also: in one's home; at home.
And Merriam-Webster has an entire article breaking down adverbs as "a most fascinating POS (part of speech)." In this system, home would be an adverb of place. Adverbs are the catch-all category of traditional grammar.
Adverbial? Prepositionless Prepositional Phrase?
Home as adverb is less satisfying to linguists who want a more nuanced description of how language functions on a syntactic level. Does home really work the exact same as other adverbs of place, which might also be considered prepositions (on, around), adjectives (backwards), or nouns (everywhere) in other contexts? More robust labeling of a word or phrase's function gets around the difficulties around parts of speech. So Rod Mitchell and other linguists can say home is a noun, and when it appears to be an adverb in the traditional system, it is actually a prepositional phrase (or in Rod Mitchell's words a "prepositionless prepositional phrase") where the preposition is elided. (The OED poses elision as a possibility in the etymology for "home, adv.") In this reading, the three sample sentences might be rendered like so, with parenthesis representing the elision:
He is (at) home ("home" is noun within a prepositional phrase where the preposition is elided)
He is at home ("at home" is a prepositional phrase denoting location)
He went (to) home ("to home" is a prepositional phrase denoting destination)
In this kind of treatment, the explicit or implied prepositional phrase might also be called an adverbial phrase (ThoughtCo), signalling that it is performing the function of modifying a verb, adjective, or clause. This kind of terminology is still generally familiar to the traditional grammar crowd, but technical enough to acknowledge how phrases and single words can both have adverbial functions.
Preposition?
Then there are readings that insist home should be another part of speech altogether - a preposition. Huddleston and Pullum have argued that home should be considered a preposition. Pullum models a more accessible version of the argument in Lingua Franca: prepositions don't merely govern/precede nouns and pronouns; they work with clauses and even by themselves. They want the class of prepositions to be expanded to include these spatial words. Maria Brenda explains in greater depth in her exhaustive book on the spatial preposition "over":
Another group of words which do not license noun phrase complements consists of spatial terms which function as a goal complement with the verbs come and go and as a locative complement of the verb be (Huddleston and Pullum 2002:614). The goal complements of the verbs mentioned, such as ashore, upstairs, home, or indoors, are traditionally considred adverbs. Huddleston and Pullum (2002) argue, however, that they should be reassigned to the prepositional category. They claim that adverbs generally cannot function as goal complements to the verbs of motion and as locative complements to the verb be.
The idea is that other spatial adverbs don't substitute in:
x 1 (modified) He is locally.
And also that the words like home are not modifying the verb, any more than young in
He is young
modifies the verb. Brenda takes issue with this last line of argument, pointing out that words like home don't modify the subject of the sentence either, and thus it seems premature to suggest they aren't in some way modifying the verb. Whether you agree with Huddleston and Pullum or not, they show how linguists are attempting to group or classify usages like home according to deep syntactic functions, and how they test each other's own terminology through comparisons and exceptions.
Best Answer
Home is a preposition according to modern grammars such as Oxford Modern English Grammar (Aarts 2011) or The Cambridge Grammar of the English Language (Huddleston & Pullum 2002).
According to such grammars, although prepositions often—prototypically in fact—take noun phrase complements, this is not an essential feature of the word category. In the same way that verbs that don't take objects are still verbs, prepositions that don't take objects are still prepositions.
First of all, notice that home patterns like other preposition phrases in the Original Poster's example, as a complement of the noun way:
In addition home can be modified by the specialised adverbs right and straight, which modify prepositions but not usually adverbs:
In addition, home cannot be modified by the adverb very which freely modifies adverbs (at least those with a gradable meaning) but not prepositions (even those which are gradable):
Unlike adverbs, prepositions such as home can function as locative complements of the verb BE:
Unlike adverbs , preposition phrases commonly modify nouns:
Thus the evidence would seem to suggest that home is a preposition and not an adverb.