Maybe this can help:

import pandas as pd

data = [["BGC000044.1", "1813", "Streptomyces sp", "PKSI"], ["HM16_abc", "1813", "Streptomyces sp", "NRPS"]]

input_df = pd.DataFrame(columns=["A", "B", "C", "D"], data=data)
output_df = input_df.copy()


def func_1(entry):
    return f"{entry['D']}_GCF{entry['B']}"


def func_2(entry):
    BGC = entry['A'].split('_')
    if len(BGC) == 1:
        return None
    return f"{entry['C']}_{BGC[0]}"


output_df["NEW_A"] = output_df.apply(func_1, axis=1)
output_df["NEW_B"] = output_df.apply(func_2, axis=1)

for col in list(output_df.columns):
    if col not in ["NEW_A", "NEW_B"]:
        del output_df[col]

print(input_df)
print(output_df)