This is a good first question and a good first coding attempt!

Layout

Conventionally the imports are placed at the top of the file, before any other statements:

import random

print(" Welcome to the number guessing game")
play_game = True

The code is very consistent in its use of whitespace (indentation, space around operators, etc.). However, there are a couple places which could be slightly improved. The black program can be used to automatically add consistency.

UX

When I run the code, I get this prompt:

Your number are between

That wording is a little awkward. Consider this instead:

print(f"Guess a number between {random_integer1} and {random_integer2}")

Note that it uses an f-string to simplify the code.

For input prompts, I like to add a single space at the end of the text so that there will be a space between the text and my input on my screen:

number = int(input("what is your guess? "))

It is great that you specifically give the user the expected options for input ("y/n"). It is also common to allow input to be case-insensitive by calling lower():

play_again = input(" Do you want to play again? (y/n) ").lower()

Ouch!

Goodbye Coward

Be nice to your users so they'll want to come back and play again :)

Simpler

Here is another print line:

print("Correct, only took you", guess_count+1, "tries!")

which is commonly written using an f-string

print(f"Correct, only took you {guess_count+1} tries!")

In this code:

if play_again == 'y':
    play_game = False
    play_game = True

the following line is not needed and should be deleted:

    play_game = False

These 2 if statements:

if number > random_integer:
    print(" Too high, guess again")
if number < random_integer:
    print(" Too low, guess again")

can be simplified as an if/else:

if number > random_integer:
    print(" Too high, guess again")
else:
    print(" Too low, guess again")

There is no need for the extra check since you already checked for equality earlier on.

The same is true here:

if number == random_integer:
    running = False
    print("Correct first try")
if number != random_integer:

This is simpler:

if number == random_integer:
    running = False
    print("Correct first try")
else:

DRY

You have duplicate code in a couple places, specifically where you ask for input, then check if the number is correct or not. Consider breaking your code into some functions for that.

Documentation

The PEP 8 style guide recommends adding a docstring at the top of the code.

"""
Number-guessing game

Here's how you play... add description here
"""