Learn Hardcore Portuguese (Portugal): O programador vai corrigindo os erros no computador com paciência. - The programmer keeps correcting the errors on the computer patiently.

corrige os erros = he corrects the errors (a general, habitual, or simple present action; also can mean he is correcting them now, depending on context).
vai corrigindo os erros = he goes on correcting / keeps (slowly) correcting the errors.

vai corrigindo (literally goes correcting) suggests:

an ongoing process;
something done gradually, step by step;
often a bit of duration in time.

So the sentence with vai corrigindo paints a picture of the programmer working through the errors, not just the bare fact that he corrects them.

Does vai corrigindo mean a future action, like he is going to correct the errors?

No. That future meaning uses a different structure:

vai corrigir os erros = he is going to correct the errors (near future).
vai corrigindo os erros = he goes on correcting / keeps correcting the errors (ongoing, gradual action, usually in the present or around now).

So:

ir + infinitive (e.g. vai corrigir) → future/near-future.
ir + gerúndio (e.g. vai corrigindo) → progressive, gradual action.

Why does this sentence use vai corrigindo instead of está a corrigir or está corrigindo?

All three express an ongoing action, but there are variety and regional differences:

está a corrigir os erros – normal, standard European Portuguese progressive form: he is correcting the errors.
está corrigindo os erros – strongly associated with Brazilian Portuguese; in European Portuguese it’s understood, but much less natural in everyday speech.
vai corrigindo os erros – common in European Portuguese to show an ongoing gradual process, often more descriptive, with the idea of bit by bit / as time passes.

So the sentence chooses vai corrigindo to emphasize the process unfolding over time, not just the simple fact that the action is in progress.

Does the -indo in corrigindo correspond directly to English -ing (correcting)?

Formally, yes: corrigir → corrigindo is the Portuguese gerund, similar to English correcting.

Functionally, though, European Portuguese does not use the gerund everywhere English uses -ing. In EP:

For the ordinary present continuous (is correcting), you usually see
estar a corrigir, not estar corrigindo.
The gerund appears more in:
- combinations like ir + gerúndio (e.g. vai corrigindo);
- some written or formal styles;
- fixed expressions (e.g. falando sério – seriously speaking).

So the form corrigindo looks like correcting, but the patterns of use differ from English.

What exactly does no computador mean here? Is it in, on, or at the computer?

Literally, no computador is em + o computador = in/on/at the computer, but in natural English you would normally say:

on the computer or
on his/the computer (depending on context).

In Portuguese, em is very flexible. Here it can mean:

he is correcting errors on the computer (e.g. files, programs stored there), or
he is correcting errors using the computer.

Context would decide which English preposition sounds best, but no computador is the standard, idiomatic way to say this in Portuguese.

Why is it no computador and not no seu computador (on his computer)?

European Portuguese often omits possessive adjectives (his, her, my, etc.) when it’s obvious who something belongs to.

So:

no computador can easily be understood as on his computer, if context makes that clear.
If you really need to specify, you can say no seu computador or no computador dele (his computer).

In many everyday sentences, especially with things clearly tied to the subject (e.g. o carro, a casa, o telemóvel), the bare noun with article is enough.

Why is it os erros and not just erros without the article?

Portuguese uses definite articles much more often than English.

os erros = the errors (either specific errors already known from context, or sometimes errors in general).
Bare erros (no article) is possible, but sounds more generic or more abstract, and in many everyday contexts it would feel incomplete.

In this sentence, os erros no computador suggests:

the particular errors that exist in that computer/program/code, rather than errors in general as a concept.

Can we say com paciência somewhere else in the sentence? Does the position change the meaning?

Yes, you can move com paciência around without changing the basic meaning. For example:

O programador vai corrigindo os erros no computador com paciência.
O programador, com paciência, vai corrigindo os erros no computador.
Com paciência, o programador vai corrigindo os erros no computador.

All mean essentially the same thing: he corrects the errors patiently.

The main effect of moving com paciência is emphasis:

At the beginning → stronger focus on with patience.
In the middle (with commas) → adds a slight pause, highlighting patience as a side remark.
At the end (as in the original) → neutral, very natural location.

Why does the sentence use com paciência instead of pacientemente?

Both are correct:

com paciência = with patience.
pacientemente = patiently.

In practice:

com paciência is more common and sounds very natural in everyday European Portuguese.
pacientemente is a bit more formal or bookish; you’d still hear it, but less in casual speech.

So the sentence chooses the more common, more neutral-sounding com paciência.

What is the gender and number pattern of programador? How would it change for a female programmer or for plurals?

Programador is masculine singular:

o programador = the (male) programmer.

Other forms:

Feminine singular: a programadora
Masculine plural: os programadores
Feminine plural: as programadoras

So, for a female programmer, you would say:

A programadora vai corrigindo os erros no computador com paciência.

The rest of the sentence stays the same.

Could we just say O programador corrige os erros no computador com paciência? How would that feel different from vai corrigindo?

Yes, that sentence is perfectly correct:

O programador corrige os erros no computador com paciência.

Difference in feel:

corrige – simple present: he corrects them (habitually, or right now, depending on context). It’s more neutral and straightforward.
vai corrigindo – adds the sense of progression, step-by-step, the action unfolding over time.

So vai corrigindo is more vivid, like watching him work through the errors, whereas corrige just states what he does.