Is this an ok type of job for immigration?

Ok, so I found this private English teaching job where the director of a women's hospital is setting up her own private English classes because she wants her friends, family and staff to better understand/speak the language.

The issue is that we signed a private contract for a year, but she isn't running a real English school or anything.

So would immigration see this as a form of illegal work?

I'm a legit teacher with qualifications but because it is a private contract with a woman that technically doesn't run a licensed English school, would that be an issue?

I want to show immigration that I have more work/money coming in but I don't want it to cause more trouble rather than help me out.

This is just a personal opinion. I could be wrong.
Based on what you said:
The issue is that we signed a private contract for a year, but she isn't running a real English school or anything.
And
The issue is that we signed a private contract for a year, but she isn't running a real English school or anything.
I think if the immigration officers find any reasons to check you up. You may end up having some troubles. Based on your description of the "job", it sounds unofficial even with the existence of the contract.
Is this a regular "job" where you work full time hours or just few hours. Do you have a regular job and this "job" is secondary?
My piece of advice to you is paying a visit to the immigration asking for confirmation if your current is legal and has no concerns.

Is she going to be your grantee?
Will she do all taxes etc for you?
I do not think this contract will be accepted, but I am not the immigration

@Justagaijin Yes, I have a part time English teaching job with a legit school, which provides me with a decent amount of hours. Immigration knows that I'm currently working at this school. They are also aware that I was working at another part time job at the same time as my current job, because I wrote down the information for both schools at the time I applied for permanent residence.

I also informed immigration that I did quit one of my part time jobs, but that I'm looking for another job. The immigration staff member was polite and kind over the phone. He told me that it isn't required that I find a full time job or a second part time job while waiting for my permanent residence results.

Of course, I do have a lot of students that I teach English to privately and now this "job" opportunity.

She, will definitely not be my guarantor as I already have one (my wife).

I will talk to her about possibly being able to handle my taxes, but even if she is willing to do that, she doesn't have a licensed school so it might be better for me to edit the contract to 6 months rather than a year and then move on to a legit job at a school.

it sounds like your wife is japanese ..

if you are on spouse visa .. you don`t need to worry about private job and immigration.

all you need worry about if your main job permit you to do extra job. and procedures of declaring your tax if you want to.

Example i am a shian in my company ... i`m not permitted to do per time job.

i