First, you need to obtain access to the New York Times data via their API. Register for an API key at the New York Times Developer Network by creating an account. Once registered, choose the API(s) relevant to the data you want to collect (e.g., Article Search API, Most Popular API) and generate your unique API key.

Use Python to make HTTP requests to the New York Times API and extract the required data. Utilize libraries such as `requests` for API calls and `json` to parse the response. Write a script to automate data extraction, ensuring you handle pagination if applicable:
```python
import requests
import json

api_key = 'your_api_key_here'
url = 'https://api.nytimes.com/svc/topstories/v2/home.json?api-key=' + api_key

response = requests.get(url)
data = response.json()
# Process and store the data as needed
```

Once the data is extracted, transform and clean it according to your requirements. This may involve parsing JSON objects, cleaning text fields, normalizing date formats, and handling any missing values. Use Python pandas for efficient data manipulation:
```python
import pandas as pd

# Assuming data['results'] is the relevant data list
df = pd.json_normalize(data['results'])
# Perform cleaning operations
df.fillna('', inplace=True)
```

Convert the cleaned data into a CSV or Parquet format, which Snowflake can easily ingest. Use pandas to write the DataFrame to a file:
```python
df.to_csv('nyt_data.csv', index=False)
```

Snowflake requires data to be loaded from a cloud storage service. Choose a cloud storage service like Amazon S3, Google Cloud Storage, or Azure Blob Storage. Upload your CSV file to a chosen bucket/container in your cloud storage. Use the respective CLI tool or web interface to upload the file:
```bash
aws s3 cp nyt_data.csv s3://your-bucket-name/
```

Log into your Snowflake account and create a table schema that matches the structure of your CSV file. Use the Snowflake web interface or SQL commands to define the table structure:
```sql
CREATE TABLE nyt_articles (
title STRING,
abstract STRING,
url STRING,
published_date DATE
-- Add other columns as necessary
);
```

Use the Snowflake `COPY INTO` command to load data from your cloud storage into your Snowflake table. Ensure you have a valid Snowflake stage that points to your cloud storage location:
```sql
COPY INTO nyt_articles
FROM 's3://your-bucket-name/nyt_data.csv'
FILE_FORMAT = (TYPE = 'CSV' FIELD_OPTIONALLY_ENCLOSED_BY = '"');
```

By following these steps, you can effectively move data from the New York Times into a Snowflake destination without relying on third-party connectors or integrations.