Let's analyze the problem step by step. We are given a step-up transformer with the following specifications:

Primary Voltage (V_p) = $220 \mathrm{~V}$

Secondary Voltage (V_s) = $11000 \mathrm{~V}$

Power (P) = $88$ watt

The power input to the transformer is equal to the power output, assuming there is no loss in the transformer.

Hence,

$$P_{primary} = P_{secondary} = P = 88 \mathrm{~W}$$

The power equations can be written as:

$$P_{primary} = V_p I_p$$

$$P_{secondary} = V_s I_s$$

Given the power in the secondary winding is $88 \mathrm{~W}$ and the secondary voltage is $11000 \mathrm{~V}$, we can find the secondary current $I_s$ using the formula:

$$P_{secondary} = V_s I_s$$

Rearranging to solve for $I_s$ gives:

$I_s = \frac{P}{V_s}$

Substituting the given values:

$$I_s = \frac{88 \mathrm{~W}}{11000 \mathrm{~V}}$$

Simplifying this expression:

$$I_s = \frac{88}{11000} \mathrm{~A}$$

$$I_s = 0.008 \mathrm{~A}$$

$I_s = 8 \mathrm{~mA}$

Therefore, the current in the secondary circuit, ignoring the power loss in the transformer, is 8 mA.

The correct option is Option A: 8 mA.