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I would be very happy to help you.  I am a private English tutor from England, but now living in Denmark.  I suggest that you try to write your letter in English first.  Then if you send it to me I will correct it and explain the grammar to you.  If that is too difficult for you, I can help you to translate your letter from Danish to English.  Please contact me if you are interested.  I offer good rates for students, especially if working via Skype and/or email.

Tror at hende dér vil have penge for det, men ved det ikke.

Jeg vil gerne hjælpe dig.

This is more Danglish than English, -it's quite hard to believe that it was written by a native speaker of English.

Simplified answer to help a student who says they are not very good at English.  Can assure everyone I am British.

Simplified answer to help a student who says (they are) she is not very good at English.  Can assure everyone I am British.

Can assure everyone, madam, that you left out the subject in your final sentence.

Uh, oh, this boads not well.

Well spotted!  Apologies to piabing for not noticing that you are a girl.  Still doesn´t change the fact that I am British!!

´bodes´ not ´boads!´

It is more correct to say:  Uh, oh, this doesn´t bode well! 

or more formally - this does not bode well!

By the way, it is perfectly acceptable to say ´they are´ if you don´t know whether the subject is male or female.

It's not a question of not noticing that piabing is a girl, but of failing to see that  "a student"  (singular) and "they are" (plural) are not in agreement.

The question of gender is not relevant here at all.

OK, so how would you write it?

I chose the form on purpose – cf. "I kid you not". Not your standard Modern English, admittedly, but quite acceptable in the proper context. As for the spelling blooper; no argument there. I guess I'm a bit tired; I've been up since 5 am today.

...........as I did in #5  "she is", -the only natural way to put it.

"they are" would refer to a third party.

I kid you not"  and "this bodes not" are both good Shakespearean English.

(Jul.Caes. I,ii,52   "the eyes sees not itself, but by reflection" +  Jul.Caes. I,ii,198  "I fear him not")

Periphrases with "do"  came into use in the early 17th cent.

Fair enough, rakijovic!

As I said above, Stygotius, where the gender is not known, we use they!

She/he/it are third person, so is ´they´ in the third person narrative.  It is incorrect to say The student said it was good at tennis, so we say The student said they were good at tennis.  If the gender is known we can say The student said he/she was good at tennis.

Good sparring with you boys, but I must go now!

The "English Teacher" makes a hash of it and has got this all wrong.

What she probably has in mind is something of the opposite kind, namely the lack of a common gender singular pronoun in English and the use of "they" to make up for this in instances like "Everyone would buy their own house if they could afford it".

The case here is very different and does not lend itself to such a construction. The only thing to do is  -as I have already mentioned-  to write "a student who says she is not very good at English"  -simple, fluent and  unproblematic English.

The examples she gives are not pertinent to this debate.

This "EnglishTeacher1" may be a good teacher of English for all I know, but I'll bet you all a fiver each that she is not a native speaker of English.

A sentence like "The student said they were good at tennis" is sheer nonsens unless the student is talking of other people, -not including him/herself. 

#0: Jeg kan godt hjælpe til. Bor selv i USA og har en del erfaring med at skrive jobansøgninger på engelsk. Hvor ligger det omtalte brev henne? 

Du tager fejl, Stygotius. Det er meget udbredt og generelt accepteret (i hvert fald på britisk engelsk) at erstatte "he or she" med pronomenet they - også efter et substantiv i singularis såsom student.

I "The Cambridge Guide to English Usage" (2004) står følgende om "singular they":

"For those listening or reading, it has become unremarkable—an element of common usage.

På oxforddictionaries.com kan man læse

You can use the plural pronouns ‘they’, ‘them’, ‘their’ etc., despite the fact that, technically, they are referring back to a singular noun:

- If your child is thinking about a gap year, they can get good advice from this website.

- A researcher has to be completely objective in their findings.

Du burde også læse denne blog blog.oxforddictionaries.com/2012/06/he-or-she-versus-they/. Her noteres det, at brugen af they efter et substantiv i singularis såsom

 If you’re allergic to oils or perfumes, remind your therapist before they lay hands on you.

på ingen måde er ny, og har været almindelig siden det 16. århundrede.

På en.wikipedia.org/wiki/Singular_they kan man læse, at brugen kan dateres helt tilbage til det 14. århunderede og blev brugt af forfattere som Chaucer, Shakespeare, Jane Austen, Thackeray, and Shaw. På wiki bliver brugen af they efter ordet student eksplicit nævnt

The singular antecedent can also be a noun such as person, patient, or student

JN12,

Det er nogle fine eksempler du har fundet frem, og naturligvis kender jeg udmærket de bøger du nævner, men det er ikke så heldigt at du svarer i vest når nogen spørger i øst.

Alle dine fine citater har ikke noget med denne sag at gøre.

Det er heller ikke særlig heldigt at du farer frem  og vil belære nogen om noget du ikke selv har styr på, navnlig ikke når du prøver at belære fagfolk. Det bliver let pinligt.

Jeg har ikke "taget fejl" af noget som helst.

Det som du og denne "EnglishTeacher" har fået galt i halsen, er brugen af flertalspronominet "they" til at referere til et pronomen i ental hvis køn man ikke kender eller som kan dække begge køn (hvilket ikke fremgår af sammenhængen)   fordi man på engelsk ikke har noget entalspronomen i fælleskøn.

Det handler dine eksempler ikke om.

Dit første eksempel har f.eks. intet med denne problemstilling at gøre.

Brugen af "they" har man som regel efter pronominer som "everyone og no-one og someone" og lign. som f.eks. "Everyone can not afford to buy a house of their own"   ....for at undgå det lidt tunge "his or her own".

Den aktuelle sætning "a student who says they are not very good at English" kan kun betyde at studenten taler om nogle andre end sig selv. På den brugte måde er det det rene nonsens.

Du og denne "EnglishTeacher" har ikke forstået denne mekanisme.