Jeg burde have hjulpet hende med flytningen i går, men jeg var på arbejde.

Because after a modal verb like burde (should / ought to), Danish normally uses an infinitive, not a finite past tense verb.

So the pattern is:

• burde + infinitive = should
• burde have + past participle = should have done

Here:

• burde = should / ought to
• have = have
• hjulpet = helped

So jeg burde have hjulpet hende means I should have helped her.

burde hjalp is not grammatical, because hjalp is a past tense form, and you do not put that directly after burde.

It expresses a past obligation, expectation, or regret about something that did not happen.

So:

• Jeg burde hjælpe hende = I should help her
• Jeg burde have hjulpet hende = I should have helped her

The second one looks back on the past. It often suggests:

• the speaker thinks helping would have been the right thing to do
• but it did not happen

In this sentence, the second clause explains why:

• men jeg var på arbejde = but I was at work

So the whole sentence has the feeling of: I know I ought to have helped, but I couldn’t because I was working.

After modal verbs in Danish, you normally use the bare infinitive without at.

Common modal verbs include:

• kan = can
• vil = will
• skal = shall / must
• må = may / must
• burde = should / ought to

So:

• jeg kan hjælpe
• jeg skal arbejde
• jeg burde have hjulpet

Not:

• jeg burde at have hjulpet ❌

And not:

• jeg burde havde hjulpet ❌

Havde is a finite past tense form, while have here is an infinitive.

Hjulpet is the past participle of hjælpe (to help).

The forms are:

• at hjælpe = to help
• hjælper = help(s) / am helping
• hjalp = helped
• hjulpet = helped

In burde have hjulpet, Danish uses:

• modal verb
• infinitive have
• past participle hjulpet

This is the same participle you also see in perfect forms such as:

• Jeg har hjulpet hende = I have helped her

Because hende is the object form of the pronoun, while hun is the subject form.

Compare:

• Hun hjalp mig. = She helped me.
  Here hun is the subject.

• Jeg hjalp hende. = I helped her.
  Here hende is the object.

In your sentence, jeg is the subject and hende is the person receiving the help, so the object form is required.

Flytning means move / moving / relocation.

So:

• en flytning = a move
• flytningen = the move

The ending -en is the attached definite article, so instead of a separate word like English the, Danish often adds it to the noun.

Here, med flytningen means with the move or more naturally with the moving.

It is definite because the speaker and listener presumably know which move is being talked about — for example, her move.

So:

• hjælpe hende med flytningen = help her with the move

Yes, often you could.

Compare:

• hjælpe hende med flytningen = help her with the move
• hjælpe hende med at flytte = help her move / help her with moving

These are very similar, but there is a small difference in feel:

• med flytningen focuses on the move as an event or task
• med at flytte focuses more directly on the action of moving

Both are natural in many contexts.

Yes, i går (yesterday) can often move around in Danish, especially in a main clause.

Here it comes after the object phrase:

• Jeg burde have hjulpet hende med flytningen i går

This is natural and clear.

You could also hear:

• Jeg burde have hjulpet hende i går med flytningen
• I går burde jeg have hjulpet hende med flytningen

These versions are also possible, though they may shift emphasis slightly.

A useful point for learners: Danish word order is somewhat flexible with time expressions, but the position of the finite verb still follows the normal main-clause rules.

Because men is a coordinating conjunction, and after a coordinating conjunction Danish normally keeps ordinary main-clause word order:

• subject + finite verb

So:

• men jeg var på arbejde = but I was at work

You would get inversion (verb before subject) after certain fronted elements, for example:

• I går var jeg på arbejde = Yesterday I was at work

But men itself does not trigger inversion.

På arbejde is a fixed idiomatic expression meaning at work.

So:

• Jeg er på arbejde = I am at work
• Jeg var på arbejde = I was at work

Danish often uses certain prepositional expressions without an article where English learners might expect one. This is one of them.

So you should learn på arbejde as a set phrase.

Both are possible in Danish in some contexts, but var is the normal simple past here.

• jeg var på arbejde = I was at work
• jeg har været på arbejde = I have been at work

In this sentence, the speaker is talking about a finished time in the past: i går. Danish very often uses the simple past for that.

So var is the most natural choice.

The first clause follows normal Danish main-clause order, where the finite verb is in the second position.

Breakdown:

• Jeg = subject
• burde = finite verb
• have hjulpet = non-finite verb phrase
• hende = object
• med flytningen = prepositional phrase
• i går = time expression

So the structure is roughly:

• Subject + finite verb + rest of verb phrase + object + other elements

This is very typical Danish word order.

Burde is completely normal and common, especially in statements about what would have been the right thing to do.

That said, Danish can also express similar ideas in other ways, for example:

• Jeg skulle have hjulpet hende ...
• Jeg havde egentlig tænkt mig at hjælpe hende ...

But burde have hjulpet is a very standard and natural way to say should have helped.

A small nuance:

• burde have often sounds more like moral correctness or personal regret
• skulle have can sometimes sound a bit more like expectation, plan, or what was supposed to happen

In many everyday cases, though, the difference is small.

Because men joins two main clauses:

• Jeg burde have hjulpet hende med flytningen i går
• jeg var på arbejde

In standard Danish punctuation, a comma is used before men when it connects two independent clauses.

So the comma helps mark the contrast clearly:

• I should have helped her yesterday,
• but I was at work.