Before beginning, familiarize yourself with the GNews API documentation to understand how to fetch articles and what data is available. Similarly, understand the data requirements and structure of Convex to ensure compatibility.

Set up your programming environment with the necessary tools, such as a text editor or an IDE (like Visual Studio Code or PyCharm), and ensure you have Python installed on your system. Install necessary libraries like `requests` for making HTTP requests.

Use Python to write a script that sends a GET request to the GNews API endpoint. Make sure to include your API key in the request header. Parse the JSON response to extract relevant data fields such as title, description, and publication date.

```python
import requests

api_key = 'YOUR_GNEWS_API_KEY'
url = 'https://gnews.io/api/v4/search?q=example&token=' + api_key

response = requests.get(url)
data = response.json()

articles = data.get('articles', [])
```

Based on Convex's data requirements, transform the fetched data into a format compatible with Convex. This may involve mapping fields from GNews to Convex, adjusting data types, or restructuring the JSON.

```python
transformed_articles = []
for article in articles:
transformed_article = {
'title': article['title'],
'description': article['description'],
'published_date': article['publishedAt']
}
transformed_articles.append(transformed_article)
```

Ensure you have access credentials for Convex and understand their API authentication method. Set up the script to authenticate your requests to Convex, utilizing tokens or keys as necessary.

```python
convex_token = 'YOUR_CONVEX_API_TOKEN'
convex_headers = {
'Authorization': f'Bearer {convex_token}',
'Content-Type': 'application/json'
}
```

Create a POST request to send the transformed data to the Convex API. Ensure that the request is correctly structured to match Convex's expected data input format.

```python
import json

convex_url = 'https://your-convex-instance.com/api/upload'
for transformed_article in transformed_articles:
response = requests.post(convex_url, headers=convex_headers, data=json.dumps(transformed_article))
if response.status_code != 200:
print(f"Failed to upload article: {transformed_article['title']}")
```

After uploading, verify that the data has been successfully transferred to Convex. Check the response from the Convex API for any errors or confirmations of success. Implement error handling in your script to manage failed uploads or data discrepancies.

```python
for transformed_article in transformed_articles:
response = requests.post(convex_url, headers=convex_headers, data=json.dumps(transformed_article))
if response.status_code == 200:
print(f"Successfully uploaded article: {transformed_article['title']}")
else:
print(f"Error uploading article: {response.text}")
```

By following these steps, you can effectively move data from GNews to Convex without relying on third-party connectors or integrations.