Here are some of the budget phones with 4G Functionality.
Redmi Note Prime
This is one of the best phone from Xiaomi with 4G functionality. It has 64-bit Qualcomm Snapdragon 410 processor with 2GB RAM. There are two versions of storage support 16GB and 32GB. The device has HD IPS display of 5.5'' inch. The device has Android Marshmellow installed on it. As for the primary camera it has 13MP which makes it a good phone for anyone with photography interest. It has 3100mAh removable battery.
The price on the official site is listed as Rs. 7999.
You can read about it here : http://www.mi.com/in/noteprime/
This model from Acer is very cost effective with impressive features. It has Mediatek MT6753 Octa-core 1.3 GHz Cortex-A53 and 3GB RAM. The device is dual SIM with 4G LTE support. And has camera of 8MP primary side. Though for camera it may not be a good option.
The price of the device is listed as Rs. 8990 on Flipkart and 9,100 Rs on Amazon.
Here's the link: https://www.amazon.in/gp/product/B01ESUFQU6/
Lenovo K6 Power
This is one good model from Lenovo at the budget price. It has Snapdragon 430, octa-core 1.4GHz processor with 3GB RAM. The device has 5 inch IPS dispaly and 13MP. As a dual SIM device with 4G support, it has some good hardware.It has Android 6.0 installed on the device. For this price the device has very good performance for 4,000mAh, non-removable battery.
the price listed as Rs. 9990 on Flipkart.
Moto E3 Power
This device from Moto has Mediatek MT6735P quad-core 1GHz Cortex A53 with 2GB RAM. And it has the display size of 5 inch. The device has 8MP primary camera with 5MP secondary camera. With storage space of 16GB and 3,500mAh, non-removable battery it has good performance. With Dual SIM slot support and 4G, this is a good device for this price range.
It is priced at 7990 Rs at Flipkart.
Lava Iris Pixel V2
Pixel V2 seems to be very good model at this price. It has 1.0 GHz quad-core, 64-bit MediaTek MT6735 processor. The device has 2 GB RAM with 5 inch display. The primary camera has 13MP with 16GB storage space. It has Android Marshmallow installed in it. With 2500 mAh Li-ion removable battery, it has a good performance for this price range.
The price listed is 7999Rs at Amazon India.
These are some of the models which are under Budget pricing and are good in terms of 4GLTE functionality. Mi and Lenovo phone is definitely worth checking if you want to buy 4G phone at budget price.