Can anyone explain the specifics on how the Hinayana Arhats are "fallible and imperfect" and "possible for to relapse?"
As has been pointed out, there were and still are differing opinions on the matter.
The Mahāsāṃghikas, from which the Mahāyāna was born, are a good example of the position that postulated the fallibility of Arhats. Mahādeva, charged with having initiated schism in the sangha, is said to have been an early advocate of the Mahāsāṃghika school. He is reported to have held five "heretical" views:
-Arhats can be led astray by others;
-Arhats are still subject to ignorance (despite their awakened state);
-Arhats are subject to doubt;
-Arhats can be taught by others (and are therefore not omniscient);
- [various forms, all revolving around the notion that] it is [somehow] permissible or good to say "Oh, the suffering!" [etc.]
Whether Mahādeva really existed and/or held such views aside, this at the very least indicates some people very early on considered Arhats to be subject to doubt and ignorance, which was enough to prompt polemics from the Sthaviravāda.
I suspect this view arose as a result of some individuals considering the Buddha as transcendental and while nominally being an Arhat, actually being well beyond Arhatship. This would mean arhatship is not the ultimate state to achieve. The Mahāsāṃghika perspective is summarized nicely by Venerable Guang Xing in his work The Concept of the Buddha
(page 53) as follows:
The Mahāsāṃghikas’ religious philosophy was based more on faith than on reason, and accepted whatever was said by the Buddha or, more precisely, whatever was taught in the Nikāyas and the Āgamas. As a result, they developed the concept of a transcendental (lokottara) Buddha based on the superhuman qualities of the Buddha, as discussed in Chapter 1 above. Two aspects of the Mahāsāṃghikas’ concept of the Buddha can be identified: the true Buddha who is omniscient and omnipotent, and the manifested forms through which he liberates sentient beings with skilful means. Shakyamuni was considered but one of these forms. The true Buddha supports the manifested forms that can appear in the worlds of the ten directions. In Mahayana Buddhism, the former aspect – the true Buddha – was developed and divided into the concept of the dharmakāya and the concept of the sambhogakāya; the latter aspect – the manifested forms – was developed into the concept of nirmaṇakāya. Thus, the Mahāsāṃghikas are the originators of the idea of the nirmaṇakāya, and the manifested forms can have many embodiments. Furthermore, they also introduced the theory of numerous Buddhas existing in other worlds.
One might gather from this perspective that there is something beyond arhatship and that arhats themselves, subject to ignorance and doubt, are clearly not perfected beings. This is of course the seeds for the early Mahāyāna which would have raised questions about whether or not sentient beings might attain the same state as the Buddha and not just mere liberation from samsara, which the Arhat is credited as having accomplished.
"possible for to relapse?"
Again, if an Arhat is said to be subject to ignorance and doubt, then presumably they might carry out karma which would initiate retrogression. In much of early Śrāvakayāna it was assumed that the driving force behind involuntary rebirth was desire (kāma), which is why it was said that lust in an Arhat is simply non-existent. From that point of view, then it is unimportant if an Arhat is still ignorant and not in possession of omniscience because the whole goal is cessation of desire. The twelve links of dependent origination are "propelled" as it were by desire.
However, some might have suggested that the real cause for rebirth was ignorance with desire being a secondary by-product of that ignorance, in which case the ignorant Arhat, despite having achieved cessation of desire, could still be subject to eventual rebirth regardless.
This is actually what much of the Mahāyāna came to insist: Arhats are reborn due to their ignorance, albeit in favourable circumstances. Not all Mahāyāna thinkers agreed with this. However, this simply has to be the case for the proponents of Ekayāna (the "one-vehicle") which insist that all beings regardless ultimately do attain unexcelled perfect enlightenment as buddhas. That means both icchantikas and arhats alike. This question is addressed by Nāgārjuna in his Mahāprājñā-pāramitôpadeśa
答曰：得阿羅漢時，三界諸漏因緣盡，更不復生三界。有淨佛土，出於三界，乃至無煩惱之名，於是國土佛所，聞《法華經》，具足佛道。如《法華經》說：「有羅漢，若不聞《法華經》，自謂得滅度；我於餘國為說是事，汝皆當作佛。 (CBETA, T25, no. 1509, p. 714, a9-15)
Question -- Arhats in their past lives must have extinguished all the conditions and conditions to receive a new body. Where do they abide and perfect the Buddha's path?
Answer -- When one attains arhatship all contaminated causes and conditions of the three realms are extinguished and one is no longer reborn in the three realms. There is a pure Buddha-land beyond the three realms, even being without the word 'defilements'. In this realm, the place of the Buddha, they hear the Lotus Sūtra, and perfect the Buddha's path. As the Lotus Sūtra says, "There are arhats who, if they have not heard the Lotus Sūtra, think of themselves as having attained cessation. In another realm I explain this - you all will become buddhas."
Arhats are then said to be reborn outside the the three realms. This is another matter that the Mahāyāna addressed: Arhats have complete knowledge of the three realms, but incomplete knowledge of things outside the three realms.
My main source of information (on how Yogacara makes the distinction) is from Tsongkhapas Ocean of Eloquence translated by Gareth Sparham.
Yogācāra does not advocate Ekayāna. They also postulate the existence of icchantikas such as in the Yogācārabhūmi-śāstra
. These are beings who can never be liberated from samsara ever. Their ideas on the eventual fate of arhats would be of a similar vein: once the causes for rebirth are severed, then one is "extinguished" forever with no possibility of being reborn or interacting with reality ever again. This is of course a reification of cause and effect, being settled in conventional truth without understanding of ultimate truth, that is easily refuted with Nāgārjuna's dialectic. Hence, Yogācāra is not true Mahāyāna. Their views are easily refuted and what we have then is the capacity for all beings to achieve buddhahood, i.e., buddha-nature.
What this means is that arhats, while fallible and perhaps subject to varying levels of regression, do not achieve absolute cessation.
This is important to note because if arhats were infallible and actually did achieve absolute cessation of existences, then there would be no ultimate truth -- there would only be conventional truth of cause and effect.