Trotzdem habe ich immer ein Hustenbonbon in der Tasche, und meine Mutter vertraut zusätzlich auf ein altes Hausmittel.

Trotzdem means nevertheless, even so, or all the same. It is an adverb.

It is not the same as obwohl:

• trotzdem = starts a new main clause
  
  • Trotzdem habe ich ...
• obwohl = introduces a subordinate clause
  
  • Obwohl ich krank bin, ...

So they can express a similar idea, but they use different grammar.

Because German main clauses usually follow the verb-second rule.

That means:

• the first position can be taken by one element, here Trotzdem
• the finite verb must come next, here habe
• then comes the subject, here ich

So:

• Trotzdem habe ich ... ✔
• Trotzdem ich habe ... ✘

This is a very common German word-order pattern.

Immer means always and is an adverb of frequency.

In this sentence, it comes after the verb and subject:

• Trotzdem habe ich immer ein Hustenbonbon ...

That is a very natural position in German. A rough pattern here is:

• connector/adverb
• finite verb
• subject
• time/frequency adverb
• object
• place

German adverb placement is somewhat flexible, but this version sounds normal and idiomatic.

Because German very often forms compound nouns by joining words together.

• Husten = cough
• Bonbon = candy / sweet

So Hustenbonbon is literally a cough candy, meaning a cough drop or lozenge.

This is extremely common in German. English often uses separate words where German uses one compound word.

In German compounds, the last part usually determines the gender.

Here the last part is Bonbon, which is neuter:

• das Bonbon

So the compound is also neuter:

• das Hustenbonbon

This same rule helps with many compound nouns.

Because Hustenbonbon is neuter, and in the accusative singular, the article for neuter stays ein.

Compare:

• masculine: einen Hund
• feminine: eine Tasche
• neuter: ein Hustenbonbon

So:

• ich habe ein Hustenbonbon ✔

The verb haben takes a direct object in the accusative, and the accusative form of ein for a neuter noun is still ein.

Because here in der Tasche describes location, not movement.

German uses:

• dative for location: in der Tasche = in the pocket/bag
• accusative for direction/movement into something: in die Tasche = into the pocket/bag

So:

• Ich habe ein Hustenbonbon in der Tasche. = it is located there
• Ich stecke ein Hustenbonbon in die Tasche. = I put it into the pocket/bag

That is a very important pattern with two-way prepositions like in.

Here auf etwas vertrauen means to rely on something or to place trust in something.

So:

• meine Mutter vertraut zusätzlich auf ein altes Hausmittel

means that the mother also relies on an old home remedy.

A useful point: vertrauen can appear in more than one pattern.

• jemandem vertrauen = to trust someone
  
  • Ich vertraue dir.
• auf etwas vertrauen = to rely on something
  
  • Ich vertraue auf dieses Mittel.

Both are common, but the structure changes.

There are two things happening here.

1. Why accusative?
Because the verb phrase is auf etwas vertrauen, and this pattern takes auf + accusative.

So:

• auf ein Hausmittel
• not auf einem Hausmittel

This is a fixed verb-preposition combination.

2. Why altes?
Because Hausmittel is neuter:

• das Hausmittel

And after ein with a singular neuter noun in the accusative, the adjective ending is -es:

• ein altes Hausmittel

So the form altes is due to neuter singular after ein.

Hausmittel means a home remedy: a traditional, simple remedy people use at home, often passed down in families.

It does not necessarily mean formal medicine from a doctor or pharmacy. It often suggests something old-fashioned or traditional, especially with:

• ein altes Hausmittel = an old home remedy

Also, like many German nouns, it is a compound:

• Haus = house/home
• Mittel = remedy/means

Zusätzlich means in addition, additionally, or also.

It adds the idea that this is one more thing the mother does:

• she does not only do something else
• she also relies on an old home remedy

Its position is fairly flexible, but here it naturally modifies vertraut auf:

• meine Mutter vertraut zusätzlich auf ein altes Hausmittel

Normally, German does not require a comma before und when it joins two main clauses.

So this would also be possible:

• Trotzdem habe ich immer ein Hustenbonbon in der Tasche und meine Mutter vertraut zusätzlich auf ein altes Hausmittel.

The comma in your sentence is best understood as optional, used to make the sentence easier to read because it joins two fairly full clauses.

So:

• with comma = acceptable
• without comma = also acceptable

It is in the present tense.

• habe = present tense of haben
• vertraut = present tense of vertrauen

In German, the present tense often covers what English expresses as:

• a general fact
• a habit
• something that regularly happens

So here it describes habitual or usual behavior:

• I always have ...
• my mother additionally relies ...