Errors:  

Facing health issues" can be expressed more naturally as "experiencing health problems."

Long walking hours" should be expressed as "extended working hours" for clarity.

Corrected version:

I am writing this letter to inform you that I am experiencing health problems due to extended working hours.

Improved version:

I am writing to inform you that I have been experiencing health problems as a result of prolonged working hours.

Errors:  

Facing a back problem" should be "experiencing a back issue" for more formal phrasing.

Corrected version:

For the past few days, I have been experiencing a back problem that is becoming more noticeable with each passing day.

Improved version:

Over the past few days, I have been experiencing a worsening back issue.

Errors:  

"Consulted to" should be "consulted with" or simply "consulted."

"Doctor" should not be capitalised unless it is used as a title.

Corrected version:

I consulted with my doctor yesterday.

Improved version:

Yesterday, I consulted with my doctor regarding this issue.

Errors:  

"Sitting on one location" should be "sitting in one location."

"For longer hours" should be "for extended periods" in British English.

Corrected version:

This problem arises due to sitting in one location for extended periods.

Improved version:

This issue has arisen from sitting in one position for prolonged periods.

Errors:  

"Doctor" should be "The doctor."

"Suggested me to have rest" should be "suggested that I rest."

"I am the only earning member" should be "I am the sole earning member."

Corrected version:

The doctor suggested that I rest, but I am the sole earning member of my family, so I cannot take complete bed rest.

Improved version:

The doctor advised me to rest, but as the sole provider for my family, I am unable to rest fully.

Errors:  

"Full time worker" should be "full-time employee."

"From last two years" should be "for the last two years."

"As data engineer" should be "as a data engineer."

Corrected version:

I have been working in your organisation as a full-time employee for the last two years as a data engineer.

Improved version:

For the past two years, I have been employed full-time in your organisation as a data engineer.

Errors:  

"Will really appreciate" should be "would greatly appreciate."

"If you allow me part time work" should be "if you could allow me to work part-time."

"For few weeks" should be "for a few weeks."

Corrected version:

I would greatly appreciate it if you could allow me to work part-time for a few weeks.

Improved version:

I would sincerely appreciate it if you could permit me to work part-time for a few weeks.

Errors:  

Additionally" should be followed by a comma.

"Week days" should be "weekdays."

"Weekend" should be "weekends."

"So that work will not hamper" is awkward and should be rephrased.

Corrected version:

Additionally, I will be available from 10:00 a.m. to 2:00 p.m. on weekdays and from 11:00 a.m. to 5:00 p.m. on weekends so that work is not disrupted, and I can also get some rest.

Improved version:

Furthermore, I will be available from 10:00 a.m. to 2:00 p.m. on weekdays and from 11:00 a.m. to 5:00 p.m. on weekends, ensuring that work is not compromised while allowing me sufficient rest.

Errors:  

Leftover work" should be "remaining tasks.

"Keep my laptop along with me" should be "keep my laptop with me."

Corrected version:

I will manage my remaining tasks at home and keep my laptop with me.

Improved version:

I will manage any remaining tasks from home and ensure I have my laptop with me at all times.

No Errors:  

Corrected version:

I look forward to your positive response.

Improved version:

I look forward to your favourable response.

Errors:  

"Your" should be capitalised.

Corrected version:

Yours sincerely,  

Shalini

Dear Sir,

I am writing to inform you that I have been experiencing health problems as a result of prolonged working hours. Over the past few days, I have been experiencing a worsening back issue. Yesterday, I consulted with my doctor regarding this matter. The issue has arisen from sitting in one position for prolonged periods.

The doctor advised me to rest, but as the sole provider for my family, I am unable to rest fully. For the past two years, I have been employed full-time in your organisation as a data engineer. I would sincerely appreciate it if you could permit me to work part-time for a few weeks.

Furthermore, I will be available from 10:00 a.m. to 2:00 p.m. on weekdays and from 11:00 a.m. to 5:00 p.m. on weekends, ensuring that work is not compromised while allowing me sufficient rest. I will manage any remaining tasks from home and ensure I have my laptop with me at all times.

I look forward to your favourable response.

Yours sincerely,  

Shalini