How would someone claim cashscript p2sh funds with their authorized address?

This is the example from p2pkh.cash showing a simple authorized address example:

pragma cashscript ^0.7.3;

contract P2PKH(bytes20 pkh) {
    // Require pk to match stored pkh and signature to match
    function spend(pubkey pk, sig s) {
        require(hash160(pk) == pkh);
        require(checkSig(s, pk));
    }
}

This contract only has one function. It checks the hash of the public key provided to spend matches the public key hash specified for the contract, then it checks that the signature is valid for that public key.

You can input it here:

https://playground.cashscript.org/

And funds the contract (through funds sent via your testnet wallet). To switch between wallet view and contract view, there is an ⇆ icon in the upper right of the playground.

There are free coins from here:

https://tbch4.googol.cash/

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The op codes in the redeem script actually being generated under the hood look like this:

<the authorized public key hash bytes20> OP_OVER OP_HASH160 OP_EQUALVERIFY OP_CHECKSIG

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If you get an error submitting the transaction, you can debug it with the meep command, if you install the meep debugger.

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If you wanted to call the spend from javascirpt, it would be something like:

const tx = await contract.functions
  .spend(myPubicKey, new SignatureTemplate(myPrivateKey))
  .to(<someaddress>, <someAmmount>)
  .send();

If you wanted to use it from another language, you could use mainnet.cash. The documentation to use it from REST is here

node p2pkh.js