How does the 2nd Noble truth explain the cause of suffering-as-suffering (e.g. resulting from a severe burn)?

The 2nd noble truth states the arising (samudhaya) of suffering is due to craving that leads to new 'becoming' or 'ego-existence' (bhava). In other words, craving without becoming does not lead to the arising of suffering.

And this, monks, is the noble truth of the origination of stress: the craving that makes for further becoming — accompanied by passion & delight, relishing now here & now there — i.e., craving for sensual pleasure, craving for becoming, craving for non-becoming.

SN 56.11

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The craving that makes for further becoming — accompanied by passion & delight, relishing now here & now there — i.e., craving for sensual pleasure, craving for becoming, craving for non-becoming: This, friend Visakha, is the origination of self-identification described by the Blessed One."

MN 44

The audio lecture 4. Noble Truth of Dukkha's Origin (part 1 | part 2) will provide a proper explanation.

The Second Noble Truth states that there is an origin of suffering and that the origin of suffering is attachment to the three kinds of desire: desire for sense pleasure (kama tanha), desire to become (bhava tanha) and desire to get rid of (vibhava tanha). This is the statement of the Second Noble Truth, the thesis, the pariyatti. This is what you contemplate: the origin of suffering is attachment to desire.

Ajahan Sumedho

Cūḷavedallasutta:

Miss, how sukkha-vedanā clearly appears as sukkhā, and dukkhā?

1. Āvuso visākā, sukkhā-vedanā clearly appears as sukkhā when it's arising, and dukkhā when it's changed (vipariṇāma;vanish).
2. Dukkhā-vedanā clearly appears as dukkhā when it's arising, and sukkhā when it's changed.
3. Adukkhamasukhā-vedanā clearly appears as sukkhā when it's arising simultaneously with vijjā, dukkhā when it's arising simultaneously with avijjā.

Note: Above sutta talking about the clearly appearing, so the answers mentioning just the adukkhamasukhā-vedanā, which harder to appear clearly than the other vedanas. But naturally, avijjā can arise with every vedanā in S.N. Saḷāyatanavagga, nirāmisasutta.

In paṭiccasamuppāda, dependent origination:

1. Every steps are arising simultaneously with phassa, vedanā, saññā, cetanā, jīvitindriya, and manasikāra in S.N. Nidānavagga, Samanupassanāsutta.
2. Also, every taint steps, craving & aversion, are arising simultaneously with avijjā, in Samanupassanāsutta, too.
3. So, caving causes the first Noble Truth, such as dukkadukkha. Because avijjā arising simultaneously with taṇhā, sukhā/dukkhā/adukkhamasukkhā vedanā, and bodily/verbal/mental apuññābhi-saṅkhāra, that causes, not simultaneously, The first Noble Truth, bad resultants (dukkha-vipāka) which arising simultaneously, such as viññāṇa, nāna&rūpa, saḷayatana, phassa, dukkhā (dukkhadukkha)/adukkhamasukkhā vedanā, etc.
4. But, ariyamagga causes the third Noble Truth. Because vijjā arising simultaneously with sukhā/adukkhamasukkhā vedanā, and bodily/verbal/mental puññābhi-saṅkhāra, that causes, not simultaneously, saupādisesanibbānadhātu, ariya's life, which leading to anupādisesanibbānadhātu, the third Noble Truth.

See the path of purification, CHAPTER XVII — THE SOIL OF UNDERSTANDING (CONCLUSION): DEPENDENT ORIGINATION, for more detail.

The "problem" might be that one is to much attached to see cause and effect as a straight linear pattern.

Suffering with suffering shows good the fact, that phenomenas are not only caused by past actions, but also how reaction is giving at the present and actually there is the door of release and the chain works here in a more subtly way, in a manifold smaller scale. If taking the raw tool, used to, to investigate, the picture is certainly blurred. That's why refined states of awarness are required, lack of clinging/maintaining (to objects of odentification) is nessesary to do not only see the cause of suffering but also the cessation as it act-ually comes into being.

Discernment: Right View, in wings of awakening, might be supportive to clear up doubts.

That is way it is good to understand depending origination as depending co-arising and to understand "from this comes that" and/or "with the arising of this, that arises".

Or let it be explained with simply words:

It’s like this knife: “I want to lift up just its blade, but the back of the blade will have to come along as well.” Learn to think in this way. Wherever there are causes, things can arise again. They won’t stop.

Still, Flowing Water

[Note: This is a gift of Dhamma, not meant for commercial use or other lower wordily gains by ways of exchange or trade]

Dukkha-dukkha is just painful feelings accompanied by craving. Painful feelings are simply feelings. In the case of severe burns, one would experience a lot of painful feelings. But it is still just a physical feeling like any other physical feeling. It becomes suffering only when you wish it to go away. That is called Vibhava-tanha. In other words, the craving for a certain experience to go away causes you suffering.